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Affiliation on a paper written mostly in previous position
If research was done and paper was mostly written at institute A, but then it finally got accepted while the author moved to institute B, say, 3 years later. Should the affiliation of the author be

Only Institute A: because 95% of the support was from here, and work was done here
Both Institute A and B: in some sense, both institutes supported the work
Only Institute B: this is where the author is affiliated at the moment

related: Changing affiliation on publication

There are no fixed rules but I would opt for your option (2). The affiliation is intended to aid in facilitating contact with the author but is of course useful to a department to show count the paper as a product from that institute. By listing your former address first indicating that that is where you did most of the work and then adding the second as present address provides the best and useful information for all parts. Option (1) means your present location is not disclosed which is a missed opportunity to locate you. Option (3) has the disadvantage that your former department are not associated with the work you performed there. So although all are acceptable, (2) would be the best (most polite and useful) way in your situation.

Is there any actual examples of using option (2)?

Firstly, some journals have specific rules about what counts as an affiliation. So if the journal has such rules you should follow them. These rules are variable, and I have seen all three of the options included. In my opinion, affiliation should match your current contact information and, on top of that, match any affiliation where you conducted the research,  if possible. Note this means options (2) is best. If the journal has a rule precluding option (2), be sure to thank any institution you don't put as an affiliation in the acknowledgment section. All universities deserve credit for what they contributed. That said, in my experience the majority of people in this situation (not a huge sample size), use the affiliation that either matches their current contact information or their contact information from when they submitted the paper. However, this doesn't mean it is what they should do.

I fully disagree! This might result in a major uproar since the funding and research opportunities were given by A.

It's one thing to down vote; it's another thing to vote to delete. Strongly disagreeing with an answer is not grounds for deleting it.

This is a good answer.

Your current institution is the one you should have most allegiance to.

Just let me clarify I did not vote to delete the answer, it is valid, but I disagree ;).

I think my previous version of this answer was being misinterpreted [mostly my fault]. I was recommending option 2, but was noting that option 3 is very common and that if you do go that route you should, at least, thank the old institution in the acknowledgements.

@OBu I don't think we disagree, please see the edited answer that more accurately reflects my views.

the current version is much clearer now and I changed my vote.


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What makes securing faculty positions difficult?
This question is an off-shoot from this one, where it has been agreed by most that securing faculty positions is difficult in general. I would like to know what exactly makes this so.
Though in theory university rankings may be pointless, there is a broad quality-based classification of institutions in any country which many will agree on - for example, the crème de la crème, top tier, middle tier and decent universities, of which there could be a few hundreds. We shall assume the student has passed out with a good thesis and impactful publications.

Is it tough for a student graduating from a higher rung to gain a position in the lower rungs?
What factors dictate the difficulty in securing a position in a university in the same league?

PS: In India, the answer to Q 1 is "not at all", as there is a heavy crunch for faculty positions even in top institutes. Instead the difficulty arises only when students from low rung colleges seek top positions: in most cases, such students are found wanting in skills.

Physics World, May 2015: only 1.7% of the current post-docs in physics in the UK will get permanent jobs (lectureship in the UK, a rough equivalent of assistant professor in the US).

One of the strange effects of faculty hiring (and graduate admissions) is that offers do not necessarily go to the strongest candidates.  Departments have limited resources to interview, recruit, and hire faculty.  Interviews are expensive; startup packages are really expensive; faculty job offers burn political capital even when they aren't accepted.
So hiring committees make strategic decisions based on the perceived probability that candidates will accept the position.  The University of Southeast North Dakota at Hoople would most likely not interview superstar applicants, because they don't want to waste their time interviewing someone who's "obviously" going to get offers from stronger schools.  As with any self-selection process, this assumption is partly justified and partly Institutional Impostor Syndrome.
So no, selecting an MIT grad is absolutely not a no-brainer for U-Cal-XYZ.
And yes, sometimes reasonable PhD students from very strong schools fail to get faculty jobs, or even interviews, because they don't quite have the research record to get an interview at the best departments, but their pedigree scares off weaker departments.

@Inquest One option is to specify in his cover letter exactly why he wants to go there.  To a small extent, this happened to me.  I wanted to go to a research school, but if I didn't get such a job, I would have preferred to go to a school near my family and where I went to school.  I said this to one less research oriented school, and was progressing through their application process until I got a few offers that I clearly preferred over them.  At that point I withdrew my application.

The short answer is that there are many more people that want faculty positions than there are positions.  Any time demand is higher than supply, cost tends to go up.  So why is demand so high? Partly it's because being a professor has lots of attractive qualities.  However, partly it's because that's what our professors tell us we should be (some do this very explicitly, and some more implicitly).  As a result, many students (especially stronger students) decide that the only way to succeed is to become a professor.
Another aspect that makes becoming a professor hard is that most professor jobs require a combination of skills: teaching, research, article writing, grant writing, advising and mentoring, networking, etc.  However, grad school generally fails to teach us many of these skills.  Most grad schools focus almost exclusively on research, and possibly teaching.

The demand-supply part is one can understand, but how do rankings and no. of univs enter this model? A top-univ graduate may easily get into any middle-tier univ, right?

To clarify: There are many more highly qualified people who want faculty positions than there are positions.

It is difficult because there are more graduating PhDs than there are faculty positions. This is because by and large, academic departments are not growing very fast.
Consider a department that has 40 faculty positions, and is not growing. Suppose each faculty member has a career that spans 40 years (Say, ages 28-68). Then in a steady state, this department will hire 1 new faculty member every year. On the other hand, say each faculty member graduates 1 student on average every three years. (This is conservative: say each professor has only 2 students at a time, and each one takes 6 years). So this department graduates 13-14 students every year.
This is what happens in general: each department produces many more PhDs than it consumes, so there must be many who leave the system.

I understand that, but can't people move down the univ. ladder? Isn't selecting an MIT grad. a no-brainer for a U-Cal-XYZ?

This happens at every level, not just at MIT. People do indeed move down, but there aren't many positions there either. Graduating from a good school of course helps your employment chances.

(This should go without saying, but the reason that graduating from MIT does not guarantee you a job at say a top 25 school is because where you graduated from is actually only a tiny part of how you are evaluated: you are primarily judged on your record of research in graduate school. Having attended a top school is correlated to having a strong research record, but not as strongly as you might think)

@^: Yes, agree with that. I have made that assumption in the question too.

One way to think about this: pedigree opens doors for you but it won't necessarily close the deal. A degree from MIT might get you more interviews - but the final decision, as JeffE indicates, is more complex.

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Applying for phd with flunked course in master
Well, here's the thing. I'm a 2nd year ME master but enrolled in a CSE course and got flunked. So up to now, I enrolled 8 course in my first year (4+4). Without the failed course, I get GPA 3.52 for the rest seven courses. My GPA drops to 3.0, if that course is included. So I plan to take one more course to balance my GPA in this semester. Actually, I feel confident about the courses in this semester.
Meanwhile, I'm gonna apply for phd in this semester. So how will the failed course affect on my application for phd? Am I supposed to submit transcript with 7 or 8 courses?

Is it even possible to get a course removed from your transcript (after you've received a grade)? I don't think my university allows such a thing, but I guess that may not be a universal quality.

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How much emphasis should be put on GitHub reputation in a computer science MSc application?
My undergraduate major was not computer science, but civil engineering, and I took five computer science courses in college. I worked as a software engineer for five years after I graduated, but not for famous companies. I love open source and I spend much time on it. Now I have a 2000+ stars GitHub open source repository and several 50-100 stars repositories.
My target is a top 30 computer science MSc program in the US.
How helpful are popular GitHub repositories like mine for applications, in particular when changing fields?

An admission committee can give you insights, but we aren't that committee, so we can't. Perhaps you can rephrase your question in terms that we can answer? (Please avoid shopping questions in any rephrasing.)

One answerable question might be how much emphasis to put on the GitHub reputation in the application.

If you don't have a top B.Sc in CS you may find this a much more significant obstacle, as you'll be competing with mainly CS B.Sc. grads with top grades.  You need to demonstrate a consistent interest in CS, not just programming.  Be prepared for extra course requirements (to ensure you are at B.Sc. CS levels) and hence extra time and extra expense.

If you don't have a top B.Sc in CS you may find this a much more significant obstacle — I doubt it, actually. Lots of people change fields between undergrad and grad school, and five years of experience and a popular github repo go a long way toward balancing out a lack of formal academic training.

2000+ stars for a GitHub repository is an amazing achievement. This will definitely make you stand out if you know how to use it right.
You should tell a story in the cover letter/statement of purpose/or whatever it is called: what is your passion with coding and open source, how did you come up with the idea for that project, how well it was received by the community (explain a bit as people may not know GitHub), and how it helped other projects (search if any other repositories that use your tool).

This part is not an answer: ask yourself why do you want a Master? This is impressive for a PhD program too. If you just want a Master, then it's better to find a job with a top tech company instead of wasting two years, and a lot of money. Stack Overflow has a function that if you give them your GitHub account URL, it can refer you to employers.

What is the function where if you give them your github, they refer you? I took a small glance and couldn't find anything like that.

@Seiyria Try the jobs board and the developer story.

@TRiG I see that and have it set up, but what I'm trying to figure out is where the "github employer match" feature is. I don't recall this existing in that capacity - just that you can put your github stuff on your profile. If there exists some documentation that SO uses it to match, that would help. Otherwise, it's just a result of having github related stuff on your profile and it being more appealing to employers (not specifically seeking to match with them).

Some countries require a master's before a PhD.

5000 stars, 20+ repos, spending years of my life in unpaid open-source work. I'm unemployed :)

It will definitely need some explanation. Many CS academics don't participate in open-source development (*raises hand*) and will have no idea what "2000 stars on github" means.

@PascLeRasc No, it's a terrible filter. The asker can continue to do open-source development in grad school regardless of what his professors do, and it's perfectly possible to have enthusiastic open-source developers on the faculty but not on the admissions committee.

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Whom to discuss research papers with?
When I read research papers I often come across many things that I'm unclear about and would like to talk over with someone. My advisor is not available to do this with me as she does not have time. I'm not sure with whom should I discuss these research papers with, in order to help me understand the papers better. I am the only student who is currently being advised by my advisor.  How should I go about finding people to talk through these things with, so I can better understand the research papers I'm reading?

What do you mean by doubts? Do you mean that suspect that there are problems with the papers or do you just mean that there are things that you are unclear about and want to talk over? I suspect it's the latter, but it's worth being clear because the answers may be different.

@BenjaminMakoHill Yes its the latter.

"Doubt" used this way is standard in Indian English, but not ordinarily used by English speakers outside India.  It's good for non-Indians to learn to understand this usage, but also for Indians to learn that it's likely to be misunderstood by non-Indians.  See http://english.stackexchange.com/questions/2429/ for more.

This distinction has caused many problems for my colleagues :)

Your advisor is not the only person to go to, to get answers to the questions that research papers are raising for you. Talk to other researchers, in your department, or online with peers at other universities.
But it does sound like you are getting insufficient advisory support. Do you really just have the one advisor? Time to build up your supervisory team.
Talk to your advisor about what's expected of you, and what's expected of them. It sounds like you've got a mismatch between your need and their resource, and it's important to get that fixed as soon as possible.
You'll also find a lot of good, relevant advice on these questions: skimming a paper and running a reading group.

How do I talk to peers at other universities. I dont know them personally. Also I dont know whether they have read that paper or not.

I'd suggest that you form a reading group! (As also suggested in passing by EnergyNumbers.)
What helped me out early during my PhD was to create a series of reading groups around literatures I wanted to learn. A model I often followed was to organize a weekly meeting to read 1 book or 3-5 papers with 2-5 other students. We'd usually meet for 2 hours or so. I found other students in my cohort/program and others in the university who had similar interests. Ask around! If the papers you are reading are the kinds of things that are likely to be on your general or qualifying exams, chances are pretty good that others around you will have to be reading them as well!

Does the person who downvoted this want to explain?

There are many things not clear in your question. For example, is the paper something that your advisor has asked you to read ? Or is it just something that you're browsing for your own edification ? Did the advisor say that she cannot or will not help, or that she's busy ?
I can understand an advisor finding it difficult to spare the time to explain papers that maybe even she hasn't read. But if it's something related to your work with her, then I'd expect her to help a little more. You have to realize though that just because someone is your advisor, it doesn't mean that they know more than you about every single topic :) - in fact, part of your evolution as a student will be to get to the point where your advisor asks you for help !
But I think the general answer is as EnergyNumbers indicates: find other students in your department to discuss these papers with. That's really the best way. Also, realize that working through a difficult paper, on your own or with others, is the best way to learn new material. It's a normal part of the training process.


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How to resolve grading dispute between TA and Lecturer?
On a math exam recently, the students were asked to use the definition of a limit of a sequence to prove that the sequence given by 3n/(3n+5) converges to 1. Given a positive number Ɛ, the definition requires proving the existence of some number N such that if n>N then |3n/(3n+5) - 1|<Ɛ.
As a consequence of the definition, once a sufficiently large N is found, any larger value of N will also suffice. Many students set |3n/(3n+5) - 1|=5/(3n+5)<Ɛ and solved for n to find N = (5-5Ɛ)/(3Ɛ). However, the professor decided to include an extra step: 5/(3n+5) < 5/n <Ɛ, which leads to another sufficient value N = 5/Ɛ.
Although most students gave a correct proof (consistent with the definition in their book), the lecturer took off points because they didn't find the "best" value of N. The lecturer claims that the author would have used some (unnecessary) inequalities to find the "better" N, which is probably true.
When students complain about losing points, I tell them that their answer is correct and that they should seek full credit for their work. The lecturer suggests that I am putting the students in a position in which they may "pick a side" and that ultimately the lecturer is in charge.
Who's wrong here?
Update: I was not notified about the lecturer's decision to remove points until after I gave the midterms back to the class. Once students started asking me about the missing points, the only written justification left by the lecturer was "not best N."
By "best N," the lecturer was referring to the N value found by using the additional inequality 5/(3n+5) < 5/n <Ɛ. By "best," he does not mean "smallest" (and by definition, there is no largest N).

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

There's a lot of confusion in the answers below: Please note that the lecturer's answer generates the larger N, as compared to the students. E.g.: for ε = 0.1, student N = 15, but lecturer N' = 50.

Is it possible that the lecturer didn't realize that his answer is not "best"? It gives a worse N than the students' N, and the proof is no more trivial than the students' proof. So what does he mean by "best"? Is there some criterion that I'm not understanding (perhaps not even imagining)?

123 says in his/her answer, "If students answer a question correctly [in mathematics] then they deserve full credit".  I'm commenting here because this attitude seems to be implicit in other answers, and perhaps in the OP too.  I strongly disagree with this opinion.  If a correct but poorly written answer, excessively complicated, hard to read and full of irrelevancies, is given full marks purely because it is "correct", this is severely unfair to a student who has taken the trouble to find a simple argument and explain it clearly.

Let's see if I got this right.  The professor graded the exams on his own?  And you weren't involved in grading them?  The two of you didn't meet to go over the official solution he was going to post?

Are you absolutely sure that the lecturer wasn't trying to claim that 5/(3Ɛ) was a slightly more elegant solution. I just cannot conceive of any mathematician claiming 5/Ɛ was best.

I believe the issue may actually be mathematical after all, even if the lecturer did a poor job at conveying this (but who knows, maybe this detail was explained carefully in lecture). As you indicate, to show that the limit of 3n/(3n+5) is 1, you argue that for any ε>0 there is an N such that if n>N then |(3n/(3n+5)) - 1|<ε. The variable ε ranges over the positive reals (and perhaps, so does N), while n should be a natural number. Now, |(3n/(3n+5)) - 1|<ε is equivalent to (5-5ε)/(3ε)<n, so you can take N=(5-5ε)/(3ε). The issue is that this quantity may be negative for large ε. (Cont.)

For small ε, this N is perfectly fine. Now, one can argue that it suffices to consider only small values of ε, but this needs to be at least mentioned. Lacking that, taking N=(5-5ε)/(3ε) has the issue that n>N does not ensure that n is positive (if ε is, for example, 23). One could then argue that this is implicit in the notation, but in truth I would expect that unless this is made explicit, at least some students would be perfectly happy with taking any n, positive or negative, such that (5-5ε)/(3ε)<n, regardless of the fact that the sequence 3n/(3n+5) is only defined for n natural. (Cont.)

So, unless some such clarification is added to the answer, there is indeed something lacking (or even incorrect) if one simply says that N=(5-5ε)/(3ε) (or, perhaps, its ceiling, if N should also be a natural) works.

Mathematics allows for objective truth. If students answer a question correctly then they deserve full credit. I do not think it is wrong for you to advocate for your students or for you to encourage them to advocate for themselves.

The whole point here is that there are multiple correct answers and the lecturer is insisting that students must give the "best" answer, without saying so in the question. An analogy would bethe question "Fire regulations require that there be at most 100 people in this room. Are the regulations satisfied?" The students have answered, "They're satisfied because there are fewer than 100 people in the room" and the lecturer is insisting that they say "They're satisfied because there are only 67 people in the room" for full credit.

"without saying so in the question." undergraduate students should know the purpose of assessment is usually a test of their understanding of the subject and shouldn't need everything spelled out to them.  Do we really need to say "show all working" for all questions we ask?  BTW I don't think the analogy works because (presumably) stating the threshold shows more understanding of the situation, rather than slightly less (as in the case in the original question).

realized latex does not work here* ... So, I will just say this: The question is about proving that a given sequence converges. Clearly, the students have done this by solving the first inequality.

This answer merely asserts an opinion. It offers no useful advice for what the OP can do in this situation. It offers no answer to the title question.

@DikranMarsupial If the students really understand the topic, they know that if you need to prove that something holds "for all large enough n" it is almost never necessary to identify the exact meaning of "large enough". So this is not testing understanding.

@DikranMarsupial : "Do we really need to say 'show all working' for all questions we ask?"  Yes!  Until you teach them otherwise.  I struggled hard for a week before mental anguish relented that "5 or -5" was a valid answer... that a math problem could have multiple answers. Maybe that is a valid assumption among higher math tiers, but college is a learning environment. Some students may not have gotten used to this yet. Furthermore, lots of students highly value grades (sometimes overly so, perhaps due to older baggage) so points off can be a very (overly?) harsh way to teach this lesson

@DavidRicherby as I said, we don't know what the lecturer has told the students in class, so perhaps the lecturer has a good reason.  You may not agree that it is a good reason, but it is generally a good idea to have some self-skepticism in this thing rather than jump to conclusions.

@DikranMarsupial It’s a fair point that the lecturer may have stated this bad requirement in the classes.

The nature of the dispute makes this problem difficult.
As a mathematics (BS) and computer science (MS, PhD) student I have done numerous exercises that required proof of the existence of a natural number N such that for all n>N some inequality is true. In addition to limits in mathematics, they show up in computational complexity analysis of algorithms.
Every time I have done one of those exercises I have picked a value of N that made the proof as simple and clear as I could. Often, I was aware of a smaller value of N that would have required a longer proof. I have never been marked down for picking an unnecessarily large value of N.
Any finite value N, no matter how large, such that the inequality is provably true for all n>N is equally good. That is an important aspect of these definitions, something the students should understand and apply.
If smallness of N were going to be a grading factor, despite its irrelevance, it should have been announced in advance.
That said, it would have been better for the OP to discuss the matter privately with the professor, and perhaps with more senior professors. The OP should not encourage protests directly, but should state the professor's decision and recommend that follow-ups be forwarded directly to the professor or offer to forward them on the students' behalf.

Of course. The smallness of the constant can't matter if you care about what's going on in the infinity. In this case the professor on the next course will now wonder why are these students doing all the unnecessary steps to find smaller constant :D

I fully agree with this. As somebody who works in this area, I've never come across a situation where it's important to find the exact cut-off for a "For all sufficiently large n"-type statement. And, even more generally, I've often been in a situation where I've just proven something strong enough for what I need, even though I know that something stronger is true. This lecturer seems to be teaching students to waste time coming up with unnecessarily precise results.

My understanding of the question is that the students picked a smaller value than the professor which works, but the justification requires extra inequalities (which maybe the students didn't explain?).

@Kimball: That's what I understood as well. Although, honestly, determining that 5/(3n+5)<Ɛ is equivalent to n>5/(3Ɛ)-5/3 is not something that should IMO require justification in a calculus course.

I've +1ed you answer, but I think it would be better to not outsource the communication aspect to another answer that disagrees with yours. Even if much of the information is the same, the fact you disagree may mean some details will differ and will probably result in a somewhat different presentation of the available options.

@Patricia: The professor didn't actually find a smaller N, but a larger one. There is a slightly larger one which simplifies the proof and only increases N a tiny bit, and the professor's change which makes N three times larger without any need.

Another way of looking at the lecturer's larger N is that it more immediately generates sequence elements that are closer to the limit of 1. (Not that it defends the terrible marking, but just an attempt at understanding the perspective.) I do recommend that Patricia edit the answer to observe that the lecturer's answer had the larger N, not vice-versa.

As a former Math / Computer Science student myself, I would like to add that some teachers use a "find the smaller possible N" clause to add artificial difficulty to their tests and to somewhat padronize the answers. I was really not satisfied with this when I was in college.

Mathematically, you are clearly right. Any reasonable person should agree with you. The problem asked to prove that a limit holds, they proved it, period. "Find the optimal N for a given epsilon" has nothing to do with the question asked[0]. Since your professor doesn't agree with you, it makes me suspect he's not a reasonable person.
Having said that, it is still annoying for him if you "go against him" by telling the students to appeal the grade (appeal which they would win, if it is done honestly). Have you ever discussed this with him prior to you discussing it with the students? What did he say?
So why don't you propose to your professor a compromise? Ask him to change the question from "prove the limit" to "find the optimal N such that this inequality holds". Or "Once you prove the limit, give an estimate of smallest N such that the error is lower than epsilon. "
You can sort of add some context to the question to make it more sensible, for example by saying that f(n) is the percentage of criminals arrested as a function of the amount of money spent, and you want to get to a certain percentage.
In short, if he wants to ask a question about the optimality of N, make him ask that question, not an unrelated one.
[0] Personally, I would argue that it is actually harmful. Understanding that any finite intervals can be ignored and that we should focus on what happens for N arbitrarily large is a crucial point to understand convergence and limit at infinity. This obsession on the exact optimal N is harmful, because it gives the impression that it matters; it would be more beneficial to instead show how a complicated inequality, for example, can be simplified by simply considering N incredibly and unreasonably big. It doesn't matter, because we are only concerned about what happens at infinity.

"Not a reasonable person..."  I agree with this.

While I agree with the beginning of your answer, I think you've misread the OP's description of the situation. As I understand it, what happened is actually the opposite of the scenario your suggestions address: the students calculated the optimal N, while the professor used a (somewhat arbitrary) simplification that yields a valid but non-optimal N, and then took points off the students for not using the same simplification.

@Ant, I wasn't notified of the scoring until students started asking me why they lost points. Since the only justification left by the grader was "find best N," I gave my honest opinion.

@TheSubstitute: With the information you gave, the professor's solution is actually not the smallest N by a long shot.

@gnasher, I never said "best" implied smallest.

@TheSubstitute Since you never said it, I'll say it now. In a problem of this sort, the smaller N is better than the larger one. The best proof might produce an unnecessarily large N, to keep the argument or the calculation simple, but that's not the best N.

@IlmariKaronen Indeed, I had misread. Thanks! :)

I think the only thing you may have done wrong is to send the students to the lecturer.  That could be (but not necessarily) construed as undermining his authority, and TA's have to watch that carefully.
But I have always instructed my TA's to advocate for the students.  I want the TA to come to me with my errors or any other problem they find.  At least once per semester I begin a lecture with, "Mr. Johnson has informed me that....and so here is what we'll do...  And I want you all to remember, when student evaluation time comes around, that Mr. Johnson advocated for you, at great personal risk to himself."  Warm fuzzies all 'round.
Anyway, I think the way to handle such things is for you yourself to debate with the lecturer.  If you lose the debate, you can tell the students that you agree with their complaint, but that you've talked to the lecturer about it and he's not changing his mind.  You might inform them of the departmental avenues for grade appeal, but advise them that such a minor issue is probably not worth it.

I'd leave out the "at great personal risk" part. As a student, I'd feel nervous about asking a TA to advocate for me if I thought the might harm their prospects.

@DavidRicherby  No.  My students have a decent sense of humor (which seems to be missing from large swaths of academics.  It's sad, really, that we've gotten to this point where we over-analyze every word in Cheka fashion.)

Tone doesn't come across well in written text. If you'd mentioned that it was in a humorous tone, I wouldn't have said anything.

+1 as this  answer focuses suggesting the TA and instructor discuss the issue since the grading of a particular problem seems to be causing confusion for several students. Perhaps the answer could be improved with adding some material @DikranMarsupial comment at top of page - (that is, there may be several things the professor is trying to accomplish at once with the grading rationale. Could be she is trying to encourage good students to go deeper and farther next time). Sure, the professor could be just being obnoxious.

Personally, I think you are right; other people who have answered think you are in the wrong. Allow me to offer some additional advice about what to do now:

It's probably not worthwhile to escalate the situation further. Probably neither of you will change the other's mind.
You might meet with your graduate director, department chair, or other person with responsibility for supervising graduate teaching in your department. Ask them what you should do in the future, when the instructor makes a decision you feel is wrong and students complain to you about it.
One possible consequence is that, in the future, you would be asked to TA under a different professor. Presumably this is a consequence which you would welcome.

And what can the students do about it?

@ClassicEndingMusic Take the class with a different teacher if they're so inclined and the school provides that option. I had that happen to me as a TA, I worked with one professor who graded his students harder/assigned harder assignments than others and a lot of them dropped the class and took it again later with an easier teacher.

@JAB But the issue here is not about the class being harder, but the grading being unfair and arbitrary, and the instructor hiding behind their nominal authority rather than acting responsibly. It is really not a comparable situation to the one you describe.

tl;dr-  You're mostly right, but it'd probably be best to approach this diplomatically.
The basic question is whether it's appropriate for you to voice your disagreement with the instructor given your role as a TA.  I'd argue that, in academia, it's entirely reasonable for you to express your disagreement; that academia isn't the place for subservient silence.

You're mostly right
It seems like we can fairly uncontroversially establish a bunch of stuff:

Mathematically, you're right.

This is mostly the course instructor's call to make.

Students who disagree with the grading policy need to speak to the course instructor.

The controversial point would seem to be whether or not you're permitted to voice disagreement with the instructor's decision.  Reasonable people may go either way on this issue.
In typical business contexts, employees are generally expected to avoid expressing disagreement with their higher-ups.  In yet more authoritarian environments, e.g. in a military chain of command, such disagreement is actively punished.
However, one of academia's core tenants is academic freedom.  It'd seem inappropriate to require an academic (like you) to not share their opinion on an academic matter (like an exam question) to students.
This can be approached diplomatically
When you share your personal opinion, you might express it as a personal perspective as an academic in the field.  This would seem well within your rights.
Then, students might ask why, if you agree with them, you don't fix it.  The simple answer is that you can't; that it's the instructor's decision, not yours.
Reasonably intelligent students will tend to understand that that means that they need to talk to the instructor without you explicitly directing them to do so.
Professional consequences
Be warned that your instructor or other job-selector may prefer to have unquestioning loyalty and may opt against giving you a position in the future, or write a weaker recommendation letter (if at all) if they're upset enough.  Standing your ground on issues like this have inherent risks.
That said, personally, I've opted to do this in the past.  When students have complained about a decision that I've disagreed with, I've bluntly told them that, yeah, the instructor's wrong, and that they'd need to take it up with the instructor since it's still their call to make.

How you frame the dispute will make all the difference: "You're wrong because xyz" is unlikely to succeed. "I'm having trouble explaining this to our students because I don't understand in the context of xyz (reasons why I think they should get credit)" is much more likely to succeed. From my experience in the military, I was never punished for a tactful question/suggestion, although my warnings were not always heeded and ultimately, as you say, the decision rests on someone else's shoulders. Dissent should be made to superiors in private; doing so in public is what gets you punished.

When I first read this question, I was astonished by the requirement to find an "optimal" N to prove convergence as it shows lack of understanding what a limit is. In my class (I did TA work) a student would get full credit even for the factorial of the reference answer.
But then I noticed that I had misread the question. Actually, the professor's N is larger than the student's so it is definitely "non-optimal". But the answer 5/Ɛ is simpler to write and to use further if it was needed.
I think there is some pedagogical value in showing that you can weaken your statements to make calculations simpler. One can find such “unnecessary” (as OP calls them) steps in many real complicated proofs. How much this knowledge should cost to the students in question is up to their professor.

You are getting two answers:

The lecturer is your superior, he makes the decisions
Mathematically you are correct

Since this is a course in mathematics, not in management, politics, or the military, he seems to me that clearly #2 is the correct answer, and that you are right.

It's a course in mathematics, but the question is about interpersonal dynamics within academia (else it would have been asked on a different stack).

-1 The mathematics (unfortunately) cannot solve an interpersonal dispute -- only people can -- and therefore this fails to answer the posted question.

The purpose of academia is to teach correctly, not to satisfy your own ego. I hope neither of you teaches, you are not good examples.

And yet, @user, you have failed to answer the question asked. The question wasn't Who is right?, the question was How do I resolve this dispute? This is an interpersonal question, not a mathematical question.

@TRiG According to your logic, no-one answered the OP's question. That's a BS comment.

I agree with many sentiments in comments/answers here, but---and I be misreading the question---my first guess from what you've said is that the students who lost points lost points for using inequalities that required justification in the professor's mind, not because they didn't use the same bound the professor did.  Does this fit in with your situation? Deducting points for incomplete justification is of course reasonable for proofs, though where to draw the line is a judgement call, and one that is left up to the professor, though you may disagree.
In any case, if you're not sure why he took off points, then you should either ask him or direct the students to.  You should never tell students to campaign for a different grading rubric.

How do you relate that interpretation to the third paragraph of the question?

@PatriciaShanahan Based on the 2nd paragraph (which indicates the professor used an extra inequality which won't give the smallest N), I think "best" here means easiest to prove, not smallest.

I think the the students used $\frac{5}{3n+5} < \epsilon$ to find that $n > \frac{5}{3 \epsilon} - \frac{5}{3}$ which leads to $N = \ceil{\frac{5}{3 \epsilon} - \frac{5}{3}}$. Instead of the preferred solution of the professor: $\frac{5}{n} < \epsilon$ to get that $N = \ceil{\frac{5}{\epsilon}}$. So even though the first $N$ is correct as well, it is not the smallest. And that's why the professor seemed to have taken points off.

@ClassicEndingMusic I'm a little confused what you mean by "the first N."  The first one mentioned in your comment in smaller than the second one (corresponding to the professor's solution).

@Kimball Yes, but to be honest I am not exactly sure either why the professor insisted on the second N. But anyways, I don't think that incomplete justification was the reason.

@ClassicEndingMusic Well, the OP is a bit vague what the students found or did---it's possible they took other values of N which were valid but their justification was not good for some reason.  I'm just saying that based on the information we have here, one shouldn't assume the professor is being unreasonable.

It's a bit difficult to answer your question because I don't find it completely clear what the point of contention is.  But reading between the lines I think I can find two.

The lecturer says "ultimately the lecturer is in charge".  He's dead right here.  You are working under his supervision.  You can discuss and disagree with his opinion, in fact you should do so (as long as it's feasible: perhaps not if there are 1000 students in the course and marks have to be absolutely definitely finalised by lunchtime).  But ultimately it's his decision.  If you're still unhappy with that decision - if you think it's mathematically and educationally wrong - then you could take the matter up with higher authority.  But this is not something you should do lightly.
The lecturer says you are enabling students to "pick a side".  He's dead wrong here.  As long as you are giving the same advice to all students in this position, you are leaving all decisions with the lecturer - which is his job anyway.  There are not two sides the students can choose between.  It rather sounds here as if the lecturer is saying "you have to support what I say because I say so" - which is unscholarly, unprofessional and unmathematical.

You didn't actually ask what you should do, but in case you want my opinion - don't do anything about the first point, unless (as I said already) you feel strongly enough to take it higher.  But I wouldn't recommend that.  About the second, I would suggest you courteously point out to the lecturer that you are not suggesting to students that their marks should be altered, but are referring them to him to make the decision, as is his right.  (And his duty - but it might be more tactful not to mention that.)
Also, keep a sense of perspective, and see if you can encourage students to do so too.  I imagine this is probably a small part of the mark for a small part of a small assignment.
For the record, I have some sympathy with the lecturer's attitude (mathematical that is - I have no sympathy with his professional attitude).  Mathematics, especially for advanced students (you didn't say what level this is) should not always marked as right or wrong and nothing else.  That said, I doubt that I would have marked the assignments as he did in this particular case.

I thinks the outcome should depend on the exact question that was asked:

if the students were required only to provide proof, which they did, they should get full credit.
if the question mentioned that the "best" value of N had to be found, and defined what was considered best, the professor is free to take off points for answers which don't meet the criteria specified in the question.

It would be inappropriate to penalize students just because they didn't guess what the professor had in mind.

Especially important in math, where answers you get are only good for what you've asked for (check the last one).


text:
How to tell a faculty member that I don't need him on my thesis committee anymore?
Out of five needed members for my doctoral thesis committee, four are fixed. For the fifth member, I approached Mr.A. He has no real connection to my research or topic but I contacted him because he was just as good a fit as anybody else in the department to be the fifth member. I have never worked with him or taken any classes but we did talk a lot in the first few years where he would just ask about my progress by the copy machine, etc. We don't even have a "rapport". The relationship is absolutely neutral, nothing positive/friendly, and nothing negative. I asked him to be on my committee and he said yes. I said I'll get back to you with dates, etc. He is a senior tenured professor.
Then about a month later, I was looking at the CV of some of the new hired (junior) faculty in the department Mr.B (this is his first year) and I see that we have a collaborator in common. Mr.B has published multiple papers with him and Mr.B is working with the same code and the code managing team that we are working with. He is definitely a good person to have on my committee. I finally see Mr.B after a few more months (because of busy schedule/travel/etc.) and he agrees to be on my committee.
The question is, how can I tell Mr.A, as tactfully as possible, after about four months of being quiet that he is basically rejected from my committee because I didn't know about Mr.B but now that I know, Mr.B is a much better match than he is? That I don't need him but I do appreciate his offer? I don't want to antagonize Mr.A or burn any bridges. Should this be in email or in person? Should I explain or just keep it terse? Should I keep it vague that I don't need him and thanks for his time? Or should I tell him about Mr.B and why he is a better match?
FYI, this was before anything official/paperwork was done so there is no problem with that.

As the answers point out, professors don't worry too much about whose committee they're on or not: fewer committees mean less work for us :). Now if A was untenured, then Ph.D committees are a (not very important) statistic of relevance in a tenure case, but this is not the case in your situation.

You mean that you are worried about giving a prof less work to do?

Directly. "I don't need you on my thesis committee any more."

@JeffE Isn't it too direct? It may be thought to be rude and have consequences for the student. Make the professor angry with him, etc.

@Parsa Of course it should be preceded and followed by more social niceties.  "Hello, Professor, I need to talk to you about something.  Prof. X recently agreed to serve on my thesis committee, which means I don't need you on my thesis committee any more. I'd be honored if you stayed on the committee, but of course I will understand if you are too busy."

Shouldn't your supervisor be communicating this? And ... what the think of the problem?

I think you're making a bigger deal about this than necessary.  You enlisted Professor A to serve on your committee because you needed a fifth man, not because of any close research connection.  More recently, but still before signing any paperwork, you found Professor B, who does have a close research connection.
Just tell Professor A that!  Problem solved.  You don't have to dither about how to present this information: send it in an email or send it in person.  The information of the previous paragraph is sufficient: you don't need to dwell on your connection with Professor A as much as you did in your question.  (But by the way: asking about your progress at the copy machine is what I would call "friendly".  At least he knows who you are and something about your progress in the program, and he cares enough to ask about it sometimes.  That puts him ahead of the curve in many academic departments at many universities.)
What you do not seem to realize is that if Professor A has no close connection with your work, it is overwhelmingly likely that he was being friendly indeed by volunteering to serve on your committee anyway, and he'll be equally happy or, more probably, a little happier not to serve.  Since he is a senior tenured professor he has lots of stuff to do.  Serving as the fifth man on a student's committee is not nearly such a prestige job that his layoff needs to be sugarcoated.  He'll understand, and he'll be especially happy that someone with more relevant expertise will be taking his place.

You can present it as a positive development that effectively does him a favor.  For example, "I'm writing to keep you up to date on my thesis committee.  I've recently discovered that one of the new junior faculty, Dr. B, is an ideal fit for my research interests, and I was hoping to add him to the committee.  I realize you are very busy, so I thought it could make sense to replace you with him before filing the official paperwork.  Does that sound reasonable to you?  Thanks again for your willingness to serve on the committee."
I can't imagine he'll object to being replaced.  If for some reason he really wants to be on the committee, you could always try to set up a six-person committee.

I did this about a year ago when I changed research tracks and advisers.
I explained the situation to Mr. C, who was on my committee and a relatively good fit about Mrs. D, who was a much better fit with my changed research trajectory. It was no problem at all.
Usually, it will not be much of a problem.

There is usually no maximum number of committee members.  Rather than removing the extra committee member, I would suggest informing him that he is welcome to continue serving on your committee, but that if he wishes to quit he may do so without causing you any problems.

I think it's better to make the decision.


text:
Are there any legal issues in having someone's book as the basis and as a textbook for an online course?
I want to create an online course and use someone's book as the basis (follow the structure, topics) and as a textbook (use exercises, examples, etc). Note that I'm not going to distribute any parts of the book to my students.
What might be the legal issues here?
How do physical universities deal with them?

Most universities have one or more people responsible for handling copyright issues. I would find who these people are at your university, and run it past them (with all the details) to make sure that everything is fine.

I cannot see any problem with this as long as no copyrighted material is accessible in such a way that it breaks the copyrights  of the book. Although the outline of a book probably can be considered the intellectual property of the author (and possibly the publisher depending on copyright), I doubt anyone would object since the purpose of writing a (course) book is to provide a product to support teaching of a specific topic. The structure of a book can often be the most logical way of presenting the material. The key issues is rather if you reproduce parts of the book such as providing the exercises on-line. As long as participants must obtain the material in the book on their own you should not worry. From this perspective it is no different from a normal university course (where it is also illegal to copy and distribute material freely from a book (or equivalent).
So for an on-line course, you just need to be very careful with reproducing text or figures from copyrighted materials in ways that breaks the rules. You can also contact the holder of the copyright (author/publisher) to obtain the right to provide parts within the course. The success of this may depend on the system you use for on-line material and what file formats you use for distributing the materials. Such details will likely be made clear by the copyright owner if you ask.


text:
How can I avoid overuse of words like "however" and "therefore" in academic writing?
I am in the field of computer science (more specifically robotics), and as I am writing, I must actively avoid using the words "however", "therefore", etc... every few sentences. This is especially an issue in proofs, as I feel like every other sentence follows from the previous.
For example, I am tempted to write something like the following:

This constraint is satisfied because... However, if blah blah..., then blah blah. Therefore, something something. However,.... (and I can go on forever).

Often, I can just delete some of the "however" or replace "therefore" with words like "consequently". I can also sometimes change the wording to use "then", "so", or "but". However (see I can't even help myself), I feel compelled to use words like "however" to help with flow.
How can I avoid using words like "however" so many times? Is this bad writing? Is this normal?

FWIW, I struggled with this in my master's thesis. Especially with 'nested' howevers (in parentheses). I found the trick of letting it rest a day or two then proofing it afresh helped, but sometimes I was just pushing the issue to the next sentence. Some topics just are complex ♂.

Not sure whether this is too obvious, but in terms of style and writing proofs, I sometimes look into papers of others (more experienced) authors. So, you can look how they avoid (or use) these words and try to get inspired.

Let me suggest an alternate view. This is in regard to writing proofs and other very technical things. You want, above all, for your intent to be clear. It may be that "however" and "therefore" are the best available words and that they clearly express the flow of the argument.
After all, if you were writing the proof purely symbolically you would "overuse" certain logic symbols in order to be exact.
And, don't let "fancy words" get in the way of your readers deep dive into the ideas themselves. Sometimes the most obvious word is the one that is most quickly understood to express your intent. This might be especially important for those who know your subject, but are not native speakers of your language.
OTOH, I haven't actually seen your prose. If an editor or reviewer objects, then you can change it to suit, of course.

Are the first words of each of your paragraphs meant to be examples?

Good catch @GoodDeeds, but no

@Buffy However, we could assume they were. Therefore, your answer is a good example.

@PierreArlaud "We could assume they were, and your answer is a good example" works just as well. Use these linking words if you need them, but they can often be removed entirely.

Agreed. When reading articles, I am often confused by the author using different words for the same things. I get like "Wait, did I miss something, isn't he talking about the same as before?". If you feel the need to use the same word a hundred times in a row, please do exactly that and not switch around for fancyness.

I find it interesting how I have never seen the abbreviation OTOH before, yet it was intuitive what it meant without even reading the rest of the sentence.

@Evorlor, OTOH I'm often confused by such abbreviations. ;-)

+1.  The one phrase I try to studiously avoid is "in other words", since if I have to rephrase it then I should just fix the initial phrasing instead.  I don't see anything necessarily wrong with using the words in the OP.

Great answer (+1), however using "OTOH" is a perfect example of a fancy expression that might confuse the reader. I am not a native speaker and don't know what it means.

@Vincent, yes, you are correct. I find the Urban Dictionary a good source for such things. There are many similar sites as well, including many dictionary sites. But the Urban Dictionary can be (very) coarse, it seems.

@Vincent it means "on the other hand".

@Evorlor i worked in a company where they were (ab)using TLAs (three-letter-acronyms) in internal wiki. They even came up with TLA index page.

Sometimes you can use "If ..., then...." or "Suppose .... Then...." // Some copy editors will have a fit if you try to use But instead of However as the first word in a sentence. // "By contrast" instead of "On the other hand." // Agree with @MaxD that technical writing should value precision over variety. Even my high school English teacher stressed that.

@BruceET: Some copy editors might even have a fit if you use However as the first word in a sentence.

@JW But that's perfectly valid English.

@wizzwizz4: I agree, as do many others (see https://www.quickanddirtytips.com/education/grammar/starting-a-sentence-with-however-right-or-wrong), but not necessarily those cleaving to Strunk & White.

It sounds to me like you're actually doing everything that you need to do already.
In fact, when you are initially writing a paper, I would suggest that you not worry about it at all. Write things as they come out most quickly and naturally, focusing only on conveying the substance of your argumentation.
Only at the very end, when you are polishing before submission, is it worth worrying about the wording. At that point, you can read the paper out loud to yourself and see if you start feeling repetitions or stumbling over sentence structure. Reading out loud is valuable because it slows you down and forces you to really hear what you have written.  If your words feel uncomfortable when you read them out loud, then fix them just like you are doing already: deleting when you can, and substituting synonyms when you can't.

+1 for reading out loud! It's an incredibly effective trick for easily spotting things that can be surprisingly tricky to spot when just reading.

(The exclamation mark is made to be read out loud, too)

ARGH! Don't substitute synonyms just to avoid repetition. If I had a nickel for every time I had to puzzle over whether an "iteration" was the same as a "step" because the author felt they were overusing one of the words, I'd never have to work again.

@Nobody Terms of art should not of course be varied, but connector prose like "however" and "therefore" most certainly can be.

I disagree with this answer. From personal experience, “we’ll worry about that later” or “we‘ll implement that properly later” is rarely seen through. If you can do it properly now, do it now (unless it really has negative impact e.g. because it would severely disrupt your flow or it’s really supposed to be a very rough draft/prototype and any additional effort would be wasted).

@Michael My observation is that it's very easy for people, especially inexperienced writers, to get stuck on wording and fail to produce the core content.

Aggressive Pruning
I agree with other answers that your repetition of however and therefore might not be a problem in this context. However, I would like to point out another option. These words are usually included as signposts for the reader, but do not change the meaning of the text. Therefore, I suggest omitting them. For example,

I agree with other answers that your repetition of however and therefore might not be a problem in this context. I would like to point out another option. These words are usually included as signposts for the reader, but do not change the meaning of the text. I suggest omitting them.

Just try removing the offending words in each sentence where you feel it might be getting repetitive. In most cases, you'll find you can remove "therefore" or "however" without impacting the argument. If the transition of ideas is jarring, leave the words in.
You mention that you remove instances like this already, but you may not be aggressive enough. In your question, you state

Often, I can just delete some of the "however" or replace "therefore" with words like "consequently". I can also sometimes change the wording to use "then", "so", or "but". However (see I can't even help myself), I feel compelled to use words like "however" to help with flow.

To me, the "however" in the last sentence is optional. You add it to emphasize contradiction with the previous statement. You could replace with

Unfortunately, I still feel compelled to use words like "however" to help with flow.

I would suggest an exercise where you remove all "however"s and "therefore"s  and then wait an hour or so. After the wait, re-read your text and re-add the words where necessary. The time gap will give you time to forget where the words originally appeared and allow you to read with a fresh perspective. This should help with more aggressive pruning.
Rephrasing
As regards to synonyms, sometimes rephrasing the sentence works better than rarely used synonyms like "ergo". For example,

I agree with other answers that your repetition of however and therefore might not be a problem in this context. I would like to point out another alternative. These words are usually included as signposts for the reader, but do not change the meaning of the text. When this is the case, I suggest omitting them.

Rephrasing can also emphasize contradiction and support in the same way that "however" and "therefore" do. For rephrasing, the same exercise as above can help, but I also find proof-readers invaluable. They often find ways of stating the same thing more succinctly and elegantly because they have a fresh approach to the text.

I'd heartily agree with the idea of trying out dropping these "connectors" entirely. The prose itself has a direction. Unless you start a new paragraph, most readers would (reasonably) assume that the next sentence follows in some way from the previous...

The example in the first paragraph is really cool (+1), but note that both of your examples from OP's question are actually about replacing rather than pruning. Sometimes there is no way of pruning without compromising on clarity.

I must actively avoid using the words "however", "therefore", etc... every few sentences

Says who? There is nothing wrong in repeating the same linking word every few sentences, in my view. Don't let the language majors guilt-trip you into thinking otherwise. That rule is way overrated.
If you are writing about matrices, you wouldn't look for synonyms to avoid repeating the word "matrix", would you? Just use the clearest and most appropriate words, and raise your threshold for how much repetition is 'unacceptable'. Unless every third word in your text is "however", I wouldn't worry.

To be fair, using the same (non-technical) word in every other sentence (barring obvious exceptions like "and", "so", "or"...) can make a paper unpleasant to read. Knowing a few synonyms (or using a thesaurus during later editing passes) is not a bad idea.

Consider the argument structure in the paper.
I find that I have the however/therefore problem when I'm writing in flow-of-consciousness narrative voice, rather than making an effort to structure my arguments.
So I write something like:

Premise P1
Therefore, Conclusion C1
However, Counterargument to C1
Premise P2
Therefore, Conclusion C2a
Therefore, Conclusion C2b
However, Counterargument to C2a
Therefore, Counterargument to C2b

Given this experience, here are some structural things I try to consider.
1) Give back-references.
Would "Therefore"/"However" be better replaced with a simplifying restatement of the point that the arguments are building upon or tearing down?

Premise P1
Given P1, Conclusion C1
Despite P1, Counterargument to C1
Premise P2
Given P2, Conclusion C2a
Given C2a, Conclusion C2b
Despite P2, Counterargument to C2a
Given C/C2a, Counterargument to C2b

2) Consider the flow of points.
Looking at the sequence of premises and arguments you have in the paper, is it the clearest way to communicate your point? Would it be better split into separate sections? Perhaps "premises/assumptions", "conclusions/inferences/extrapolations", and "counterarguments" sections?

Premises

Premise P1
Premise P2

Inferences

Given P1, Conclusion C1
Given P2, Conclusion C2a
Given C2a, Conclusion C2b

Counterarguments

Despite P1, Counterargument to C1
Despite P2, Counterargument to C2a
Given C/C2a, Counterargument to C2b

3) Maintain a consistent direction or thrust.
Heavy use of "however" may indicate that you're regularly flipflopping between each side of an argument, rather than presenting one side in full, then presenting the other in full.
4) Reserve them for building up or tearing down a point.
If you're using "therefore" in a way that doesn't build upon prior information to form a further conclusion, or "however" in a way that doesn't present a counterpoint, then examine why you're using it.
Compare this, which neither builds on, not provides a counterpoint, but appears to do both:

We gathered the data.
However, this was not easy, as we were in the field.
Therefore, we only took a few readings.

to this, which avoids that appearance:

We gathered only limited data, due to fieldwork limitations.

5) Don't thesaurize.
Contrary to other advice, I wouldn't advise trying to conceal this issue by changing the words to synonymous terms.
You can't build a good wood-framed house without knowing exactly what the solid foundational posts are, the exact position of all the beams from foundations to lintel, and exactly how the joints tie each one to another.
You can't write a good paper without knowing exactly what the solid foundational premises are, the exact position of all your conclusions from premises to final, and exactly how the arguments tie each one to another.
That means knowing when you really mean to use "therefore", rather than spackling over all your joints to hide them behind weak weasel-word phrases like "and", "so", "then", "but", "yet", "though".
Like a good joiner makes the joints a visible feature of their work, make those words a feature of your writing, calling out its structure.
6) OK, maybe sometimes thesaurize, but deliberately.
This is an edge case, but perhaps worth mentioning.
Sometimes, we use different bullets at different levels, to avoid confusion:

blah

blah
blah

blah

In a similar way, it can be useful to thesaurize in order to separate subarguments from the main argument flow. Be careful, it can end up a mess, but it's worth trying:

Premise P1
Therefore, from P1, Conclusion C1

Note as an aside that tangential premise Pt1
And so tangential conclusion Ct1
But tangential counterpoint to Ct1

Therefore, from C1, Conclusion C2
However, Counterargument to C2

It's almost always better to slice the tangent off into another section, an infobox, or even leave it out completely, though.

Both are perfectly normal in academic writing. You can mix things up by using:
However

nevertheless
nonetheless
X notwithstanding
This is not always/seldom/never the case for...
..., yet, ...

Therefore

Thus
Ergo
Hence
Accordingly
For this/that reason

Also see http://www.phrasebank.manchester.ac.uk/ for more options to express this and other common ideas in academic writing.

To be honest if I saw 'Ergo' in a text I would think the author is trying way too hard to sound fancy.

@AccidentalTaylorExpansion Cogito for this/that reason sum.

@AccidentalTaylorExpansion: Or had watched The Matrix Reloaded too often.

Very often, you use a "however" because you're describing some developing process of thought. Thoughts change or turn to something else, and this reflects on earlier assertions. But a written document is not a speech - your text does not have to be chronological relative to your thought processes (certainly not relative to your original thought processes). You can also use structure and visual appearance in your writing, which an oral presentation can't have.
Let's take your example:

This constraint is satisfied because... However, if blah blah..., then blah blah.

You could make it:

Let us now consider the whatever constraint:

blah1 : The constraint is satisfied because ...
blah2 : blah blah

This is usually a good idea. But, as other answers point out, sometimes the "however" makes perfect sense and the narrative sounds reasonable/engaging/exciting with it in place.

I like this a lot. The "looping" linear line of thought needs not be reproduced in the text which can be better structured (although I don't quite understand your example here).

EDIT: My answer here does not directly address your problem, but I see it is useful to look for synonyms sometimes, at least we avoid to use same words repeatedly.
I would suggest to consider this website to find synonyms (www.thesaurus.com). For example, I have looked for synonyms of "therefore".

As you can see in the results, there are a number of synonyms, sorted by relevance. You can check the meaning of each word by clicking on it. You can also see examples showing how the words are used in sentences.

Using a thesaurus for academic writing is a very bad idea. If the author is not sufficiently familiar with alternative word choices to pick them on their own, it is likely that the reader is not overly familiar with them either. Academic English is supposed to be very simple.

I mean the writer should not choose like a machine. Other dictionaries (e.g., OALD) are essential to check the meaning of the word before using in a particular context.

@Arno: It often happens to me that I have to look up "simple words" and then remember, hey, there is this simple world I am familiar with. Definitely words I would understand when I reas them.

@Arno "Using a thesaurus for academic writing is a very bad idea" I don't think so.

Blindly using a thesaurus for academic writing is a very bad idea, but it's not because academic writing should be simple. It's because synonyms have similar meanings, but not exactly the same meaning, and academic writing should be precise. You shouldn't use a synonym to avoid repetition; you should use a synonym iff it more precisely matches the meaning you want.

Consider also using synonyms that aren't wordy; when "accordingly" means the same as "for this reason", using the latter just adds --words that don't contribute to meaning-- meaningless words.

This is an answer that could be applied to almost any question about writing style, and may get dinged for that, but I'm going to add it anyway: Look for examples to follow. When you read papers, take note of ones that are a pleasure to read, and then read them again to see how they do it. People have given some good answers with examples, and I think they are helpful, but they are made-up examples. It's really valuable to see how actual problems of exposition have been solved in ways that are clear and satisfying to you. Maybe they turn out to use "however" rather frequently, and maybe they don't. You will learn from what you see.

In writing as with all performances: That which looks easiest is in fact the most accomplished.

In other answers, people have suggested using synonyms, rephrasing, or omitting the conjunctions. I would like to add some extra insight on this.
I believe that “however” and “therefore” should be treated differently.
“However” serves the important purpose of preparing the reader for a contradiction or a problem that will arise following the conjunction. Therefore, it is hard to omit. The solution for avoiding the overuse of “however” is to use synonyms or to rephrase the sentence.
As for “therefore”, it shows that the next phrase or sentence follows from the previous one, and it can often be omitted. Of course, the other options are also possible.
Example
I will illustrate with a random example I came up with:

The constraint is satisfied because the Σ-value is 1.5. However, we need to be careful since the standard deviation was quite high. Therefore, the experiment needs to be repeated for improved reliability.

I recommend repeating the experiment while changing the value of Ψ to 3 so that the results will be compatible. However, setting Ψ to 3 might affect the consistency of the results due to Ω being 5. Therefore, I also recommend changing Ω to 4. This solves the consistency problem, however, precautions should be taken since this value for Ω is quite low.

Now, I will rewrite this text without using “however” and “therefore” at all. You don’t need to go this far in reality.

The constraint is satisfied because the Σ-value is 1.5. Although this is within the acceptable range, we need to be careful since the standard deviation was quite high. For this reason, the experiment needs to be repeated for improved reliability.

I recommend repeating the experiment while changing the value of Ψ to 3 so that the results will be compatible. This introduces a new problem, where setting Ψ to 3 might affect the consistency of the results due to Ω being 5. Since this may undermine the experiment, I also recommend changing Ω to 4. This solves the consistency problem, but precautions should be taken since this value for Ω is quite low.

An option is to use grammar checking tools. Most LaTeX editors do not include this, but you could, for example, copy & paste a paragraph into a word processor and see what synonyms it suggests.
I also have good experience with Grammarly, but it isn't cheap and may not be worth it. But it both tries to tell you when your text is repetitive, and you can click words to see synonyms. Just really make sure that you do not blindly accept and suggestions, as it sometimes wants to replace technical terms with something that's no synonym in that context. On the other hand, it works well with LaTeX, which is a big plus.
In the end, I think some online thesaurus and word processor features are suitable for native speakers. Non-native speakers may consider using a more costly tool, especially when your reviewers notice that you're not a native speaker.

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text:
Why do so many PhDs choose to join academia instead of industry given the pay in academia is considerably lower and the workload is much heavier?
In the computer science field, an entry-level software engineer in top-tier tech companies (e.g. Google, Facebook, etc.) could earn as much as a 20-year experience professor. And the workload in academia is way heavier.
For example, my friends in Google work 8 hours a day and have the weekend while my Ph.D. friends have to work at least 60 hours a week if they want to have a good publication record. They almost stay in the lab all day long and don't have the weekend.
And also, given my observation, getting a position at an even mediocre university is even much harder than getting a position at Google level companies, let alone a Ph.D. would take more time on their education.
Why do so many PhDs still choose to be a professor while they have the choice to go to the industry?

Note that many PhD students don't work more than 60 hours per week, but have a reasonable workload. On the contrary, many people in industry work much more than 8 hours per day.

Because value can't be measured in money? If wealth is the ultimate motivation for you, then this won't make sense. But for many people it is not.

What do you actually want to achieve in life? To answer your question, different people choose different directions because they have different goals. Personally, I have never regretted going into industry after getting my CS PhD, not because of the money but because I am much happier with the work (and in 40 years I've NEVER been able to go home after 8 hours work!)

Considering that PhD is degree primarily designed for an academic career, it is not that surprising that at least some of the degree holders end up in academia.

"For example, my friends in Google work 8 hours a day" this may be the case at Google, but it is not always the case at other top tier tech companies. For example, Amazon and FB are notorious for having no work life balance at all.

In my experience workloads in Academia tend to be a lot lighter than in industry once you get passed the initial postdoc and early faculty phase.

cuz baby boomers are working later and later into their lives and not giving up jobs...

Simple example: a prof is in function 20 years, and supervises 5 PhD candidates at the same time, for four year terms. That's 25 candidates supervised by one prof, only one of which who can take the prof's job once the old prof retires. So if all 25, or even 'many' as you say, vie for the same job, only one can make it. (Disregarding other academic positions such as researchers, but in general there are fewer places the higher up the food chain you go.) Most people having finished a PhD therefore end up outside academia.

Academics have massively more free time than people who work in industry, and I don't know where you get the idea that the workload is heavier. That's a pretty fanciful statement.

There is more truth in this movie than you'd initially credit: "They expect results!"

The work-life balance situation is country-specific. In Finland, the average professor works hundreds of hours more per year than the average full-time employee, because labor laws are enforced more strictly in the industry. Working hours in the US academia seem similar to Finland, but there may be industries where people work longer hours.

Where else are you going to do research? You can get a job in industry writing code, building actuarial models, etc.; but if you want to do actual research--- not R&D for a product, but basic research--- you have extraordinarily few options outside academia.

@MattSamuel There might be examples for both scenarios. I have been told by several professors that one should do 70-80 hours workweeks.

In CS field, an entry-level software engineer in top-tier tech
  companies (e.g. Google, Facebook, etc.) could earn as much as a
  20-year experience professor....Why do so many PhDs still choose to be a professor while they have the choice to go to the industry?

First, your assumption is wrong. Most PhDs end up in industry. I don't have any source but I think this is from 90 - 99%.
If you limit your question to the 1% - 10% that become professors, they are all very successful, i.e. they have plenty of papers, promising research direction, strong network etc etc. And there is a reason for their success: they have passion, and when you have passion, money is likely not the most important thing in life.
Except for machine learning, in most areas in CS, you need to stay in academia to do research. And there are many benefits that you can only have when working in academia.

You take credit for what you have done. Products in industry are developed by a large team, and nobody can take full credit for it. But researchers can take full credit for what they do in their papers.
Reputation: you are invited to give talk, become program committee members, etc etc, and everybody will know you. I would be excited to meet an author whose paper I have read. I'm not excited at all to meet a Google employee. (I'm living in Mountain View, a small city with 80,000 residents, but more than 20,000 Google employees)
Do interesting jobs. You always work on new things in research, while the majority of tasks of a software engineer are maintenance, fix bugs etc.

(I'm a software engineer if you are curious)

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Sure, but a large chunk of the 90% have passion too. In my field, pure math, nearly everyone goes into the program with the intention of going into academia. There are far fewer jobs available than PhDs; and at the highest levels, success is a crap shoot: it depends on what contacts you make, how successful your unpredictable research projects wind up being, etc.

I'd also add that plenty of researchers & professor end up doing consultant jobs too. When a company needs know how in advanced stuff they will want to have someone that knows well the field to consult during the design&testing phases at least. Consultants are usually pretty expensive so a good professor doing just a couple of things like this per year will earn more than an engineer (yes engineers could do this to, except for their contract that probably forbids them from doing so and the lack of time)

Not exactly what you're looking for, but this science article says that whole over 50% of postdocs think they'll stay in academia, only ~20% do. The vast majority go to either industry to "other", which I suspect includes what many of us would consider industry.

I would like though to meet some google employees like Rob Pike or Ken_Thompson. Not to mention IBM employees or other companies. Academia and industry are closely related, wouldn't survive without each other.

As far as I see it, the main advantage that academia has over industry is freedom. In industry, you generally work on what your employer tells you to work on. As a professor, even at the entry level (assistant professor), you have quite a bit of leeway to work on what interests you, with the constraint that you have to find an agency that will give you money for at least some of it. Many postdocs and even some graduate students have the ability to come up with their own project ideas and pursue them, as long as they are somewhat consistent with their mentors' funding streams. The constraint of being able to find funding is a big one, but I still think that I have much more freedom than my friends in industry. As a professor, I only talk to my boss about what I'm doing once or twice per year. In academia, usually your boss is happy as long as you bring in grants.
Another reason to stay in academia is inertia. Once you've spent 5 years getting a PhD, you know a lot about what a career in academia looks like and have resources to help you move forward. Finding how your skills might fit in industry is less obvious and you might not know how to start.

I agree that freedom to pursue own research interests is grater in academia. But there is also quite some path dependcy. Within political science, for example, I can't work on political theory during my postdoc and then decide to apply for TT positions in international relations. -- And then there are also trends, fads, and potentially interesting problems and methods that unfortunately fell out of fashion.

With the constraint that you have to find someone to pay you to do it or it keeps your boss happy, is really the same side thing both sides of the fence. Plenty of non-academic roles that academics take up have a degree of freedom to direct what they entail. Plenty of academics are 'stuck' doing the thing they can get funding for.

required: http://phdcomics.com/comics/archive.php?comicid=1436

(I had a much longer answer, but realized that it could be condensed.  I also noticed your specific field of interest, Computer Science, which allowed further compression.)
Computer science != programming.
Thinking otherwise is an overwhelmingly common error, especially in industry.  If you study computer science and that is actually what you want to do, it is very difficult to find a position in industry.  If you study computer science and want to write programs, that is a very easy position to find in industry, mostly because of the error I mentioned above.

And to further this, programming != software development

I wish I could just "chose to be a professor". But you are correct in your assessment that, for people who chose to stay in academia, money could not be the motivation.
But, doing a PhD is not just obtaining another diploma, just another step in one's education. The knowledge (i.e. collection of facts related to your field of study) obtained during a PhD is so specific that it is not easily transferable. The transferable skills a PhD candidate trains for have to do with understanding scientific literature, being able to make connections, conclusions and get new ideas from it. After obtaining their PhD, one should be able to do that reasonably well with literature from any (sub)field related to theirs, after a short period of time needed to come up-to-speed with a new topic.
On the other hand, skills required for a software engineer are a different set of skills entirely. A masters education will sometimes familiarize you with the basic tools (programming languages) and concepts needed, but you eventually need to develop skills related to writing legible, repeatable, reusable code, which you mostly obtain by a lot of practice. Judging by some of the industry interviews I had after my PhD (in parallel with the postdoc interviews), I would've probably done much better in them before my PhD, or even before finishing my masters.
So, they're jobs requiring different skillsets. I chose Computer Science because it fulfilled me, unlike e.g. medicine or law. Taking this further, I want a research job because it gives me a sense of fulfilment I wouldn't get from a software engineering position. Pure research positions in top-tier companies are quite competitive, so there's a slim chance somebody who can't succeed obtaining a permanent faculty positions will be able to secure a research position in industry.
Some of the following points are what is important to me, rather what might be important to everybody, but I heard most of them echoed back to me from my colleagues in academia. Comparing research positions in industry and academia (though my experience in industry is limited to what I got from a few interviews, and then a few months where I was forced to take an industry job while waiting for my immigration documents):

Most research positions are much more flexible with working hours. People understand that, doing creative work, some days you just can't get anywhere, and some days you're on a streak and don't want to stop after 10 hours. People that are really not morning people can occasionally arrive late for lunch, and nobody says a word.
In industry, I've heard arriving to work at 9:30AM is considered as having "flexible work hours".
Even in case of very strong research ties to industry, holding an academic position allows you to chose the problems you want to work on (possibly amongst several industry collaborations, but the choice is still made according to your research interests and research group).
In industry, you need to work on the problems as dictated by the market, interesting be damned in favour of profitable.
In my academic interviews, people were interested in the problems I was tackling.
In industry, people were interested in my problem solving skills. Nobody cared about what I applied them to, just whether I can apply it to the problems they would present me with.
Academic jobs are some of the most travelled jobs out there. During my PhD, I wouldn't be surprised if the amount the University covered for my travels would have easily closed the wage gap to industry. Conferences and professional visits are an integral part of academic research.
Travelling with industry always has a promotional purpose.
I feel free to discuss my ideas with anybody, anywhere. I reach best conclusions through discussion. The thought of chatting about a topic, having an idea and having to bite my tongue is a bit terrifying to me. The worst that could happen is somebody poaching an undeveloped an untested idea and researching it themselves - not a terrible loss since I don't get to develop all of my ideas anyway.
In industry, one needs to constantly think of confidentiality, and which details of their work they are allowed to disclose.
The goal of publications, as the "tangible products" of academic research, is to share your ideas, results and findings with the world. I enjoy the idea that my work is public and adding to the collective of human knowledge (let's not discuss paywalls right now...).
In industry, all the ideas have to be intellectually protected before publishing: patented or whatever. The goal of publishing is not, precisely, to share your approach with the public. It is again promotional; to boast about the results you obtained with your new method; the details of which you might (need to) try and keep vague.

In my head, the list of contrasts goes on. It's little things and big things, but all in all, all the freedoms that staying in academic research allows me are worth more than the difference in monetary compensation industry could offer. If somebody wanted me to work a 9-5 job (or generously allowed me to work 9:30 till 5:30), working on their problems which I find marginally interesting, and not discuss my work with anybody not on their payroll (and sometimes not even that if employees are too competitive for promotions), they would have to offer me much more than the industry standards. Since I'm not a research rock star and nobody is going to offer me that, I guess I'll stay in adacemia if I can, and leave an industry engineering job as a fall-back option.

I wanted to write my own response but I realized it had too much in common with this one, so I'd rather comment. These are pretty much my reasons. I'd also add that scientific output tends to have a much larger shelf-life than whatever one is producing (if at all!) in the industry (including code).

A few other points.

Not everyone has an enormous industry to fall back on. The body of your post mentions CS, but your title doesn't limit to science fields. There is no high-paying industry for most people in the humanities, subfields of linguistics, psychology, imaging.
Personally, I'm glad I don't work for a private company and "produce value for shareholders."
Many universities are still good places to work in terms of benefits; good retirement, medical, and other perks, especially in public schools that can offer government benefits. My university has a pretty generous vacation schedule, even if a lot of people don't take their days.

I'm not in CS, but that sounds reasonable

Because all our role models are in academia, and academia is what we know.
A Bachelor student, Master student, or PhD student will be working in a university environment.  He or she will be exposed to successful researchers, internally within their university, externally when they are visiting, or when he or she is visiting a conference or even reading papers.  The power of role models is very large.  This reinforces the selection bias.  When all the role models are within academia, academia becomes the known option.  Industry?  I don't know anybody in industry, they don't go to conferences and they don't even have employee websites, only vague and general LinkedIn profiles.  They probably never come and give guest lectures.  They might visit a careers fair at my uni, but hey, I'm a postdoc and already have a job, why should I go to a careers fair, those are for undergrads, right?  In addition to what WaterMolecule and James Fennell have pointed out, this inertia is not only a matter of sticking with what we have: it is a matter of sticking with what we know: to remain in the protective cocoon of Academia.

Source: PhD Comics

I think that most of the existing answers have a flaw: they list out some rational pros of academia versus industry, assume that PhD graduates act completely rationally, and then conclude that many stay in academia for the reasons listed.
From my experience at least (mathematics PhD, now software engineer) this is not how it works. I think a huge factor in PhDs staying in academia is simply inertia. At every stage in the career process of an academic, staying in academia is the short term easiest option. When I was graduating from my PhD, it would have taken me two or three weeks to prepare post-doc applications and submit them; applying for industry jobs took three to four months of full time re-training and the significant risk that it actually wouldn't work. In the long term, for me, it was the much better option - but at the point where I had to choose, I was tempted to take the short term easy way out.
In mathematics nowadays, most PhD graduates can expect to have two or more post-docs. During this time they will have very little financial and geographic security (you have to move to where the one post-doc you were accepted for is). Foreigners in the United States, like me, also have very little immigration security during this time, whereas good private sector employers will have green cards for their foreign PhD-graduate employees within 2-3 years. Post-doc salaries are about half those in industry for recently graduated PhDs.
The fact that, despite this chronic situation in the jobs market, many PhDs continue in academia is a strong indicator in my opinion that rationality is not the main determinant.

Socially acceptable answers:

Becoming a professor is unlikely to be the goal of the majority of Ph.D. students.
Can demand higher salary upon joining company
To pursue their own interests and passion without pressure from company
To teach others and continue to learn.

Socially unacceptable answers:

Pathway to immigration into wealthy, Western societies. Count the number of Chinese, Indian and Iranian Ph.D. students in your research field and their overwhelming white, Anglo research advisors and calculate this ratio.
Pathway into high paying companies, especially by people who did not study a lucrative major. I cannot tell you how many people I know from civil engineering or chemical engineering have used their Ph.D. as a way to learn advanced software courses and join Google or a bank afterward.
They cannot cut it in the competitive industry world. Industry routinely does their own research but they usually hire the really big names: inventors, people who have written books, etc.

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How does one use a PhD "to learn advanced software courses"?

@MartinF A lot of universities offer courses in scientific computing, AI/ML, modeling, etc. at the graduate level and you don't have to be part of the major to take them. Contrast with undergraduate programs where you might have pre-reqs to take the same classes.

@MartinF What do you mean, you just take a course. That's it. Again, I cannot tell you how many people who are studying biochem or doing some stuff like finite-element modeling are taking algorithms, software engineering, machine learning and other computing courses on the side, with the firm intention of joining a software firm afterwards and never looking back

The two main reasons I can think of (thought of at the time this was relevant to me):

Momentum:

You've stamped your name a bit of the world.
You've got to grips with a lot of things unique to this world. Things that might not be valued elsewhere.
You know you are good at what you doing.

You don't necessarily want to give that up and jump to something that's realistically a very different skill set.
There are some roads industry doesn't go down:
Unless there is a perceived short-term, competitive, advantage in knowing the answer to a question, it's hard to get industry to take the question seriously.
This won't affect everyone. Lot's of interesting questions do have answers that are competitive advantages. But there is a reason academics are often considered not to be 'down-to-earth": if it's a down to earth question you want to answer, you can normally get someone to pay you answer it better elsewhere than in academia, so you do tend to leave. If not... you have your answer.

As far as I know in the majority of the cases (not all, of course) people stay in academia without being so deep in reasoning about comparisons between academia and industry.
They studied there, they had many friend around them, perceived professors as bosses and, when they find a way to begin that career, they do it.
Moreover very often families are proud of them for being “university researcher” and money is no a major problem in the initial period.
So they stay there and are very happy of this.
A part this category, few people really make a choice and have precise targets in minds; another few people come back to academia after some years of industry; another few people begin to teach back to academia after many years of industry having in mind the idea to boost the preparation of student in a way that industry need, and so on.
As a rule the percentage of “stay in academia” is about 1-2%, the percentage of target minded less than 0.1%, that of “back in academia” again 1% or less, that of “teach back” something around 0.01% or less. These figures depend on your country as well.
(Btw: I’ve been part of “teach back” people, even if, at the moment, I’m mainly deeply involved in industry projects again and do not teach or research for university anymore).

Actually, my family might be "proud to the outside" for me being a "university researcher", but privately keep asking me when am I finally going to get out of school, and why don't I go work where they pay me what I deserve (mind you, I'm fine financially). Also, there actually used to be quite a bit of chatter about academia vs. industry where I was doing my PhD, so while they might not have gone "deep in reasoning", I know a lot of my peers from my PhD at least considered it.

The pay ain't that much lower in academia as compared to industry in all countries. In several places you can earn say 70-90% of what you would as a junior "newly baked" M.Sc.
And then spend perhaps 4 paid hours per day doing what you love most in the world instead of hmm what do I know..., fix someone elses bug (who quit for a long time ago) or writing soul-numbing requirements.

@MorganRodgers that's not what I wrote. I wrote get the time do do the funny part (aka research) of the job 4 hours a day.