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What are some tools to effectively synthesize information from research?
Does anyone know any good, effective tool that people use to synthesize large amount of information from research?
By "tool" I am referring to softwares or practices that effective researchers use to:
1. retain/recall large amount of research information,
2. progress quickly into a completely new field,
3. find parallels between different concepts and fields
At the moment, I am basically going through papers one page at a time at a time, jogging down the keywords, trying my best to understand their derivation and result before proceeding to the next.
There are several flaws with my current methodology.

I write everything on paper, and paper pile up plus it is exhausting to write things down by hand
I tend to lose track of prior results as I progress deeper into a subject, I would like some sort of review.
I keep coming back to the same concept but only defined in slightly different contexts
I keep all the papers I find in one spot but have no way to reference them using any other tools, this means in a
few month I will have to trace back to the authors who wrote this or
that

I would like something to help me become more effective at quickly gather new information but at the moment I am hesitant to compile everything under the sun in a Microsoft Word or type everything in Latex but if that is the state of the art in the research community then I will begin my research compilation today!
Don't hesitate to share your ideas :) I will place a bounty on this question

Are you a Ph.D. student?  If so, I feel your pain.  Have you tried Evernote?  It can do many different things --

I have heard of Evernote but I am not sure of how common this tool is used in higher learning. I think this is relevant for anyone of all level who is interested in an area that is not so well established/cohesive i.e. no standard textbook, or fields of application, a field connected by a dozen of papers (or even ppts) under different authors

I think many people in higher ed use Evernote.  I'm not sure what you're asking for.

There are certainly tools to compile large amounts of information.  That is, record things together in some form.  Notebooks, databases, etc.  However, I believe that the synthesis of the information can not be delegated to any instrument other than the human brain. I'm sure there are people that do research as to how the brain does it. Personally, my process is to dive into details, pull back and generalize, and see where the loose ends go. Later something will look familiar, and it may take a while for your brain to figure out what connects to what, but it usually gets there.

I've become a great fan of Mendeley and tend to advocate it a lot - its a reference manager, but it also has a far more valuable property: manageable full text search in your entire library.
Yes, Adobe Reader can search through multiple documents but the output is horrendous with multiple windows and Adobe freezing (unless they fixed whatever caused it in their security settings...)
Mendeley: Search -> get a list, open documents in tabs.
Effectively what it boils down to:

reference management
library management

(I've not found something easier or more convenient than Mendeley - though I would really like an independent library free open source library management system on my Linux Home Server... - just haven't found anything that fits my taste yet - e.g. no Webserver needed, no database storage...)
As to managing the output you produce:

Structure your information

If you have data files, use meaningful folders, use something organised and systematic, write up important information.
I tend to scribble a lot, later scan and dispose of the scribbles as they seldom are looked at again, but everybody works differently. Some people prefer to type up all information immediately, the advantage of that is that the document becomes searchable.
But most importantly: Anything that is important, write it up and BACKUP.
Ideally have a backup of everything, but most definitely have at least one or better multiple backups of your important data.

For Qualitative coding and synthesizing of information, there are number of tools you can use (specifically addressing your concerns 1 & 3):
NVivo

NVivo is a powerful workspace for your qualitative analysis and mixed
  methods research offering tools so you can deeply analyze your
  unstructured data.

Leximancer

Leximancer automatically analyses your text documents to identify the
  high level concepts in your text documents, delivering the key ideas
  and actionable insights you need with powerful interactive
  visualisations and data exports.

There are many other tools, but the above two are fairly standard practice among PhD/Researchers. Also, needless to say, they will require significant investment of time (if not financial - given most universities will have institutional subscription) and quite a lot of practice in order to use them appropriately.

You might look at Docear, which is a suite of tools centered on mind maps (tree-structured topic relationship charts).  It doesn't completely address all your needs in an automated fashion.  You would still need to establish a work flow and certain habits of organizing and reviewing your notes in order to build and maintain coherent "big picture".  For example, you might create one mind map per academic article, and then create other mind maps centered on specific themes that link to the paper-centered maps.
For me, tree-structured mind maps were too restrictive.  I tend to think in terms of concept graphs and/or argumentation graphs.  Alas, there are no general purpose academic research tools featuring those types of structures and visualizations.

You mention not wanting to "compile everything under the sun in a Microsoft Word or type everything in Latex", but I think doing something similar to this may be helpful.
Here's what I have started doing and it's been very helpful for me over the past year or so as I've been learning a new field:
I am a PhD student in biology and am an aspiring teuthologist (the study of octopuses, squids, cuttlefishes, etc.). I have started to jot down everything I want to remember from the papers I've read. Not every detail, but just the things I want to have in my working memory and be able to recall in the future.
The key for me is organization so that I don't just have endless bullet points, or paragraphs. I use LyX and break up my thoughts into Parts, Chapters, Sections, Subsections, Subsubsections, etc.
Everything  I want to remember in my field has to have a place. If I learn something completely novel, I just create a new heading and file that thought in place. I'm writing it akin to a book or journal article, so every fact or idea has a reference where I got it from. I also screenshot good figures and add them.
Here's my table of contents as it currently stands to give you an idea of where I am about a year in:
Table of Contents
It sounds like you may be in mathematics so it may take some creativity to see a parallel.
What I find so helpful about this approach is:

Each new bit of information I learn and want to remember fits into an organized framework.
When I learn something new, I'm forced to think through how it fits into my existing knowledge framework when I decide where to write about it.
Because it's broken down so finely, I can easily find information I want to brush up on and only read a paragraph or two to recollect all my past knowledge that has gotten fuzzy.
Because it's written by me, my brain often slides back into the clarity I had when writing a particular section. Especially helpful when I'm looking back at a topic I haven't examined in awhile and everything is foggy.
I've chosen what I want to remember so I am never filtering through irrelevant details.
Everything has a reference attached, so if I want those details I know which paper to find them in.
As I'm adding new information in a section, I'm forced to interact with relevant past knowledge, which helps integrate ideas and synthesize main concepts.
If a colleague or future student wants to learn something quickly, I can give them this "Reader's Digest" version.
I can create a section and leave it empty to remind myself I want to learn about that topic in the future

For full disclosure: the downsides

Organization is required. You can't get too lazy and just throw info anywhere.
Sometimes I have to put the same information in multiple places from different perspectives. e.g. if I learn about octopus brain physiology, then that info could go under the octopus biology heading with a subsection on brain physiology or under the brain physiology heading with a subsection on octopuses. Sometimes this is helpful though, because I see how the same info tells me something different from different perspectives.

Since this post is getting too long for anybody to bother reading, I think I'll stop it here. I hope someone else finds this method as helpful as I do! Plus maybe one day you will have a whole book written and you can publish it!

Thanks for your help. It has been a year since I asked this question and I am just starting to compile things in Latex. It is a chore, as it delays moving on to new fields as fast as possible, but I think it would be worth it as it finally dawned on me that my research field is not as organized as it had existed in my head when I first started

Interesting question.
While no software trumps the human brain at this this type of synthesis (so take your fish oil and keep your brain healthy!), I think that you may find that Evernote greatly aids in the task (and there are similar advanced note apps like Nevernote and OneNote which provide similar functionality). This is probably not a silver bullet and is best used within a larger ecosystem of research/study aids, such as a good citation manager like Zotero and a good information search portal like Google Scholar, Pub Med, DeepDyve, etc.
I recommend Evernote because it allows me to:

quickly compile information into notes (this goes with your #1)
see it's automatically generated suggestions of which prior notes are related to the one I'm creating or reading (this goes with your #3)
easily integrate information from other sources like webpages, documents, emails, etc (this goes with your #1)
save my notes locally and/or synchronize them across devices via "the cloud" (this goes with your #1)
encrypt sensitive information but optionally leave it unencrypted for the period of time that I'm logged in (this is nice, but not directly related to your 3 criteria, I suppose)
easily organize notes, notebooks and stacks of notebooks on the fly ((this goes with your #2, I think)
easily search all notes, any given notebook or stack of notes, or easily create/search based on tags

Note also that Endnote's free version is fully functional in every way that's relevant. They have a paid version that allows you to have more that 1 account on the machine, but no research-related tools are disabled in the free version.

I'd always be weary of relying on a third party service that in theory could go under at a moments notice - ideally whatever system you use should be as self reliant as possible.

They are completely self-reliant. Hence local storage (#4) and Nevernote (main paragraph)

Evernote relies on its web access though.

No, that's not accurate. Evernote does not need web access. Evernote gives an option for cloud storage (i.e. web accesses needed) and/or offline local storage (i.e. no web access needed).


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Competitions in the applications for postdoc positions (Applied Math)
I am a fifth-year Ph.D candidate in Applied Mathematics (my general research interests include probability theory & stochastic processes & PDEs) at a research university in U.S. and I have applied for almost 80 postdoc positions (via Mathjobs.com and Interfolio) in U.S. and Europe, and up to now I just received one offer from another research university in U.S. (although I received four interviewers in total up to now, but three of them eventually rejected my applications). Actually I thought my publication record is not bad (2 published papers and 3 pre-prints at the time when I applied for postdoc positions), of course I am aware of some "super-star" applicants but I also believe that super-star applicants are relatively rare, I am just wondering how on earth the competitions for postdoc positions (not even tenure-track positions) can become so fierce? Or does such fierce competition only occur in the area of (Applied) Mathematics?

Yes, it is fierce. I doubt that you are unique. Keep trying. Try to get some local advice on your application materials and make sure you are getting excellent LoR. Disruptions in the academic marketplace occur periodically and we are in a time of uncertainty.

@Buffy Thank you for your encouragement professor Buffy. Yet it is still so heartbreaking to see that even competitions for non-tenure-track positions such as postdoctoral positions can be such fierce...

This level of competition is nothing especially new.  Coming out of the top applied math program twenty years ago, I applied for over a hundred post-doc positions.  I got one job offer, which I took, and I have had a reasonably successful career since then.

@Buzz It's amazing that (20 years ago!) you applied 100+ postdoc positions and just got 1 offer

At one point in Fall 2020, I was thinking that one had to be at least a fringe candidate for a Fields Medal to be hired into a permanent position that year.  It didn't turn out quite that bad, but it was close.

@AlexanderWoo That was absolutely INSANE!

Literature scholars have started openly discussing which areas of research will have to be abandoned because universities are not maintaining enough positions for all the areas currently being researched to continue being researched.  Poets and novelists from the 17th and 18th century whose works are studied now will be forgotten because universities are not hiring enough English professors for research on them to continue even at a minimal level.

The competition is fierce. Many PhDs get on the market with few permanent jobs. Therefore, postdoc positions are quite competitive. You applied to almost 80 postdoc positions. For how many did you contact someone who will support you locally? A postdoc is more independent than a PhD, but you are not always given "free slate" and sometimes a supervisor is required. Secondly, applying to more focused postdocs might help, especially if you fit the profile. There again, you should contact the eventual supervisor directly and do not be afraid to extend the field of your research topics.

@BeniBogosel Thank you professor! I actually contacted many faculties with whom I very interested in working. But not all of them replied to my emails, plus many of them do not know me (or my Ph.D advisor) personally. As a postdoc I guess I "have to" embrace new research directions as otherwise what you can do will be so narrow and limited.

Instead of applying via mathjobs or interfolio, can you apply directly to the professors? It could be that your applications are not being viewed by them.

As a fellow applied mathematician, I wanted to share some anecdata: every postdoc position I've held from 2014-2022 (3 held and 1 declined) has been a result of networking rather than applying to an ad.

Can always just go to wall street...

How we got here is an interesting question, but it is likely a coming together of several things. Uncertainty usually puts a damper on hiring, and it might also cause a jam up in getting employed, with many people graduating earlier still in the pool.
COVID didn't help, of course, nor did the current political situation in the US. Too many states have decided that providing tax revenue for public goods is politically impossible and so funding is poor. Universities themselves have become political targets, further affecting funding. Science itself is under political attack.
There aren't a lot of "moonshot" goals at the moment that push money into academia as there have occasionally been in the past.
I'll guess that the biggest block at the moment is just uncertainty and that may settle out (or not) once we learn how COVID will actually play out. Unrealistic views and propaganda aren't helping, and the disease isn't going to cooperate.
I, too, graduated (long ago) when the academic job market collapsed. After a huge funding boost for the original moonshot, it completely dried up overnight leaving lots of us about to earn doctorates in an empty market. Many mathematicians with doctorates had no opportunities beyond pumping gas (and, yes, people still did that for you).
Look at all your options, including some you haven't ever considered. You've been thinking about research at an R1, most likely. But there are other satisfying academic positions that can give you a good career or serve as a bridge to better times.  Industrial research labs also do some interesting things in at least a few places. If you take something less than ideal and want to move up, make sure that you develop and keep collaborative relationships and do what research/writing you are able to manage. It may turn around, but don't assume that it will be soon.

I feel that around the level of post-doc, the hiring question also shifts a bit from how good you are to how well do you fit in the group. People might of course still hire a super-star candidate, should one show up, but those are rare and tend to only apply to specific places anyway.
In general, the choice will be between several roughly even candidates. And all people involved know the vagaries of the business well enough to realize that this decision does not really boil down to numbers. For example, far too often, the number of papers you have at this stage of your career depends less on how good you are than on how lucky you have been with the review process.
So people will look less at how many publications you have, or how good the respective journals are, but at their content. Does it show mathematical proficiency and some new ideas or is it just standard methods applied to a slight variation of a standard problem? Does it fit to what the rest of the group is doing? And if it doesn't fit, can it be turned into or combined with something else so that it does?
Your task as an applicant is to convince the people who are hiring of that. Sometimes this is easy. You might already work on precisely the same problems, they know you and you know them. But considering how few people tend to work in each sub-sub-sub-topic, this is rare. Much more likely is that you are only adjacent at most and as it turns out are a dozen others. So show them that there is a connection.
When I was involved in hiring a post-doc the last time, I found it really confusing how little time most people spent on customizing their application to the position. Everyone mentions the research they'd like to do, usually in continuation of what they already did before. But while this is nice, what people care about is what research you like to do with them and how the stuff you did could be combined with theirs.
I am not saying that you should write a whole new research proposal, in fact you should keep it short, not be too specific and keep it in a prominent position so that people will actually read it. Put a few paragraphs close to the beginning of the cover-letter, where for example you mention that you did X, the people you apply to recently did Y and that you think that combining the ideas on X and Y could lead to a paper on Z. There is no need for detail here, if they want to know about X, they can read your paper, they know about Y and the whole idea is to get them thinking about the possibilities of Z on their own, which gets them to invite you for an interview to discuss this some more.
I am not necessarily claiming that you did not do this, but I've noticed that far too many people are just content with just changing recipient and date in their generic cover letter. And if you say that you already applied for 80 positions, I am not sure that you spent the necessary half a day or so for each of them to skim a few of their recent works and rewrite your cover letter.

Thank you for your answer. I did customized my applications to the positions I applied (a bit, but not a lot). Maybe it is just a matter of luck and fitness.

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What exactly does one do in a Phd from literature reviews to dissertations?
I've already posted two questions regarding this(here and here). I've gotten excellent answers but I'm still confused and this question, I believe will answer all of my doubts.
Could someone who has done a PhD or is doing it, share his/her experience? Give an overview of your full experience from the first day till the last day.  What did you do in your first year, how did you choose your topic, how your advisor helped you when you got stuck, how doing the Phd developed your problem-solving skills, etc.  Please include technical terms when they help you get a point across. For example, if your PhD was related to building a quantum computer, include terms like qubits, etc.
Basically, just walk me through your PhD, so that I get a good idea what exactly one does in a PhD, what sorts of problems does one encounter, and how your advisor helped you.

Sorry, I'm afraid it's impossible to answer this in the scope of a Stack Exchange answer! It would take an entire book. In fact, here is one such book - and it's a free ebook, enjoy :)

(Also note that everyone has a different PhD experience - due to variation between educational systems in different countries or differences between academic disciplines, due to different advising styles, and due to differences in personal characteristics. My experience, for example, was nothing like the one described in the book referenced above, and I share some of these concerns about that book.)

I think I can relate.  When I was in fourth grade, a NASA engineer visited my class and gave a slide show presentation about his work.  In the Q & A, I attempted to find out what a typical day at work looked like for him, but I had some trouble articulating my question.  It came out, "What do you do every day?"  The engineer looked a little exasperated and sad, gestured at the screen, and said, "What I've just explained."  So I will propose an edit to your question, in an attempt to get it re-opened.  See what you think.

@ff524 +1 for recommending me that book. I've read his book. The insight I got by reading his book is that you do most of the work by yourself with little or no help from your advisors and this type of research can be done at home or in industry by collaborating with professors you know(where you did your undergrad or other professionals who know you well) and publish your results without enrolling in a PhD program.

@GaryGrey It seems like you're determined to believe that the experience of a PhD can be replicated at home, by yourself.  As I said, everyone has a different PhD experience  the author of that book felt that his advisors offered little or no help; my experience has been the opposite. The author of that book also does not say anywhere in that book that he thinks he could have achieved the same outside of a PhD program; the author benefited in many ways from being in a formal PhD program, which may be difficult to recognize until one tries going the [tag:independent-researcher] route!

Also see: Is it possible to replicate the experience gained in a grad school on my own without attending one? and Are there any aspects of PhD work that do not translate well to distance learning? (many of those also apply to self-training) and What careers not requiring an academic degree might be open to a self-learner who wants to do research on pure mathematics?.

@ff524 I understand what you mean, I can't replicate what a student learns in a Phd program at home, by myself. And I'm not interested in publishing papers in journals or in conferences. What I was looking for in his book was that, how much effort he put in to do research and how much help he got when we was doing research. And it looks like he did 99% of the work by himself. This leads me to believe that if got stuck on a startup problem and if it involves research, I now hope that it is very likely that I can solve it by myself. I hope, I'm right? Thanks. :) Maybe, I'm next Elon Musk. :)

@ff524 Am I right?

If you're saying that PhD students don't get much help from advisors, other faculty, and peers when doing research, then no, I don't think you're right (and I think you underestimate the help that the author of that book got). I also think that the problems you're likely to face as a startup founder are probably not going to be "research" problems in the sense that academics use the word. If you don't want to be a professional researcher, by all means don't get a PhD.

If you want to get a better sense of what research is and whether it's something you want to do, I suggest you try to get involved in undergraduate research at the university where you're currently a student.

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Will a PhD help me as a startup founder?
I'm planning to get double honours in CS and Pure Math. (Link to my previous post.) But I'm very confused about getting a PhD. Will it help me as a startup founder?
The only thing that is stopping me from getting a PhD is time. I have excellent grades(almost perfect GPA). I'm very motivated to launch startup after graduating(maybe after working for a year or so). The only thing that's worrying me is that if I get stuck on a problem while working at my startup and the problem involves some sort of research, Will I be able to do research and solve that problem with my background(CS and Pure Math)? Say that problem is related to machine learning and involves creating a more intelligent machine.

There's also a Startups stack exchange that might have information for you.

If you're serious about starting a successful company in a CS/Math area, you should be reasonably confident that what you are doing requires no further original research.  Even successful researchers in these areas fail at most of the problems they attempt.  It's also possible, if there isn't a known good reason to the contrary, that even approximating a solution to the problem you need solved is actually (provably!) NP-hard, in which case you are stuck for good.  Brin and Page came up with PageRank BEFORE starting Google.  Most startups operate very far from the frontiers of research.

This is going to be a wild guess: In a startup, maybe it would be to your benefit to be as young as possible, to withstand the ridiculously long hours?  I have never worked in a startup, so maybe my image of them is wrong.

Check this article http://blogs.nature.com/naturejobs/2016/09/05/why-building-a-start-up-is-probably-your-most-sensible-career-path/

You can hire academics to conduct research for you, rather than doing it yourself.

@AlexanderWoo though theory is beautiful and powerful, sometimes intuitive heuristics can go a long way in real-world scenarios (even if the approximation is NP-hard).

I think @Bitwises' point is important. Not being too charmed by beautiful theory insisting on doing everything theoretically correct or fancy to be able to step down and do "dirty" or "ugly" approximations.

No.
Disclaimer: I am an academic. I have zero experience with startups.

A PhD can be a great way to develop cutting edge ideas, but a startup is not the place to conduct research, it's the place to translate proven research into a business. Major conceptual roadblocks should be sorted out and methods proven before you start a company. Consider the difference between Google as a startup and Theranos. Google had a working research artifact that they needed to develop into a commercial platform. Theranos is apparently going to implode because their basic concept has never been proven even after $400 million dollars of funding.
Startups generally aren't founded on an amazing technical idea. A well-motivated undergrad could build the basic technology platforms behind successful startups like Facebook (databases), WhatsApp (sockets programming), or SnapChat (databases plus sockets programming). There are definitely some startups that are successful because they have a great technical edge that no one else has figured out (see again Google for an example), but they're probably the exception. The bigger value of many tech startups is the fact that they serve an unrecognized need. Their future value comes from their mature systems engineering and user base. For example, Twitter serves about 350,000 tweets per minute, which is not a trivial system to create. WhatsApp and SnapChat have a user base that makes them incredibly valuable.
Research conducted in a PhD program tends to be "basic research". Research conducted for business purposes tends to be "translational research" or "development". They are related but different things.
A PhD program is probably not going to afford you the opportunity to work on a project of your choosing related to your business.
If you are able to work on business-related research there will be a minefield of legal questions over the ownership of the intellectual property when you want to monetize your venture. Unless very specific conditions are met and agreements are made before you start your program, it is guaranteed that the university or your adviser will have some claim to the intellectual property that results from your degree. This isn't necessarily a deal breaker, but it's probably beyond you if you don't currently have the resources to hire a lawyer. And you're not going to get the university to sign away its rights.
There are huge opportunity costs to getting a PhD. You will only have time for a business alongside a PhD if you are extremely dedicated. It is not uncommon for PhD students to work 50-60 hour weeks. If you work on federal research your stipend will be around $25,000-$30,000 per year, which you can easily double as an undergrad doing low-level CS stuff at a company. If you are a hard worker in the right place you could easily triple that number after the five or six years it would take you to get your degree.

Great answer (+1), but just out of interest, could you add a brief remark about what "culture" you are basing your observations upon? I posted an answer with my observations from Germany, and curiously, some of your points seem to be in direct opposition to mine.

Only a note to add - one might consider Stephen Wolfram as an exception. He created the basis of Wolfram Mathematica during his years at Caltech, which became a successful business. Unfortunately, he is not working anymore in theoretical physics.

I did not mean to construe it as though a PhD does not give you an advantage in business. There are definitely great businesses that come out of academic departments. Wolfram Research and Google are great examples of this. However, there are better ways to spend your time if your goal is "business".

@O.R.Mapper As above, my experience is as an academic in the US. My graduate department was unusually entrepreneurial, with about 1/3 of the faculty spending some of their time starting or running a business. I think the difference in our two answers is this: You illuminate the many advantages that having a PhD can grant you when starting a business. I tackle the question of whether someone who wants to go into business should get a PhD. If OP's only concern is whether they should learn how to do research before starting a business, my suspicion is that the PhD won't help them.

It's impossible to say whether you personally getting a PhD will help you personally as an entrepreneur, but I'll give you some anecdotal insight into the level of educational attainment typical of Silicon Valley startups.
I'm unaware of any research of the success rate of startups and associated founder educational attainment. So - I expect anecdotal evidence is the best we can do.
I have worked in 5 Silicon Valley startups, and am familiar with many more from tiny two person shops to billion dollar unicorns.
In Silicon Valley, it is unusual for startup founders and early employees to have PhDs.
Of the 5 startups I have personally been involved with around 1 in 10 founders has had a PhD.  Of those 5 companies, 1 grew directly out of academic research, 1 had some relation to academic research, and the other 3 had nothing to do with academic research.
I would estimate the percentage of early employees in startups I am familiar with having a PhD at 5-10%.
You can get a feel for this by looking at lists the startups funded by a various VCs, and then looking on LinkedIn to see the educational attainment of their founders:
Here is Andreessen Horowitz's portfolio of startups:
https://a16z.com/portfolio/
Let's take a look at a few of their companies and founders highest educational attainment:

Actifio - MS
Airbnb - BS, BFA, BFA
Asana - Both founders are somewhat unclear, either no degree or a BS/A
Box - No degree, BS
Buzzfeed - No degree

Ditto Apple, Microsoft, and Facebook.  Not a PhD in the bunch and I think the only one with a BS is Woz.

You mention in your post "settling" for anecdotes, but I just want to emphasize caution over looking at the educational attainment of the founders of five startups and drawing conclusions from that.

Will it help me as a startup founder?

I am going to consider this question as "Will doing a PhD help me as a startup founder, as opposed to having any other job?", based upon what I have observed with various startup-founding colleagues during my doctoral candidacy in Germany1.
One way how it might help is by giving you an opportunity to build up a start-up while working towards your PhD. This only works with PhDs that you can get in such a way that you have an income to cover your living costs (e.g. when doing the PhD is connected to a job at the university, or similar).
If that precondition is met, however, here is how the opportunity works out:

Working in research often allows you to manage your time very flexibly. Evidently, this is beneficial for preparing another project such as a start-up at the same time. (That does not mean you'll spend less time working as such than in a job in the industry. However, it can mean that you'll have the chance to attend a talk about founding startups in the middle of the day and come back to work later, or possibly even as a part of your working time if your supervisor thinks the knowledge gathered at that talk is also beneficial for the university department.)
It depends on the person, but some PhD supervisors are quite eager to help their advisees with endeavours such as building a start-up, certainly more so than commercial employers who couldn't care less about their employees' spare time activities and may even be opposed to the possible transfer of knowledge to a possible future competitor.
While working in research, chances are you have some influences on the direction your research goes towards, making it possible to somewhat align your research and startup topics. In most other jobs, you do not have much choice about the tasks you get assigned.
Unlike commercial entities, universities are often not that "protectionist" about their resources. In fact, transfer of acquired skills and knowledge into startups is a part of their mission. Hence, obstacles imposed by the university for using some research results will probably be much less of an issue compared to obstacles from a commercial employer.
The university brings you in touch with plenty of people with similar interests as yours, both among your fellow PhD candidates/colleagues and among the students you might be teaching. That is an excellent opportunity to find partners for jointly founding the startup, get in touch with who might become your first employees a few years later, or even just to discuss and exchange ideas.

1: Note that when we hear "startup" over here, we do not think "Google, Facebook, the next breakthrough product that will change our daily lives, the first step to what will soon become a multinational corporation". Rather, we think "a small consultancy that does contract work for a special niche subject, which might grow to a size of 10 to 50 employees in the next 5 to 10 years, another one among thousands".

You totally dont need a phd for starting a company. You need

friends
a vision
enthusiasm
a cheap place to sleep

A guy makes an interesting point comparing advantages and disadvantages

Given that most of the answers ignore seemingly related and important aspects of your question, I'll provide another view on it:

Will it help me as a startup founder?

It depends, of course.
Reasons why it might not:

you will mostly be preoccupied with your research for which there is only a slight chance it might be connected to the focus of your potential startup
it might be better for you to devote the time you would spent on getting a Phd to the startup directly

Reasons why it might:

potential ideas -- as you gain knowledge and expertise in an area, you might recognize a potential problem that could lead to a startup
startup culture -- depending on what your startup is doing, you might need more experts which you might be able to attract easier if you have a Phd
connections -- grad schools might be a good place to get them

Will I be able to do research and solve that problem with my
  background(CS and Pure Math)?

You will definitely more likely be able to recognize and solve a particular problem if it's in the area in which you are pursuing your Phd in. However, related to the previous question, it is less likely you will have time to do your research and apply it. (It's doable, but very hard.)

The only thing that is stopping me from getting a PhD is time. I have
  excellent grades(almost perfect GPA).

This is perhaps a minor thing, but just to mention that having good grades might be required for doing a Phd (at a particular place), it is by no means sufficient for a successful Phd. I suggest deciding to do a Phd only if you are familiar and fond of all (or most of) the aspects about it.

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text:
Ask the author of a paper for suggestions of where to publish replication results
I based my thesis on a published paper and ask its original author if he was interested in seeing the results. After approaching him (with your help provided: How to address an author showing replicated results from his paper? ) he replied

Your application of the XXX method looks very good.  Congratulations,
  and thank you for sharing your work.

and made some recommendations about the bibliography.
Now, should I end this conversation with a thank you or can I insist a little bit more and ask him if he thinks it can be published somewhere or even if he can suggest where could I publish.
(It's a bachelor thesis so it's not something groundbreaking, it's more of a replication paper).

Related: I've reproduced a paper: where shall I share the code and findings?

And as usual the answer to "can I ask [XYZ]" is "sure, why not?".

Scientific method is based on the idea of reproducibility. So being able to replicate experimental findings, or not replicate them, is an important contribution to your community. Of course it would be more groundbreaking if you were NOT able to replicate results, as that would be a sign that there is more to learn. But even a mere replication using independent data would be make a solid paper. And of course, it matters all the most if the original paper found something really exciting.
It is a bit different for "theoretical" papers as putting forward ideas when somebody else did it already, does not add much to the community understanding. Unless you derived the same conclusions starting from very different assumptions, or following a very different path, replicating theoretical ideas typically does not warrant a publication.
As for what journal, again it depends on whether you are using independent data/though process, and how relevant was your original result. You could aim for the same journal of the original paper, and if that fails, aim for something just slightly lower in ranking, and more field-specific. Good luck!


text:
How to revise/resubmit a rejected article taking an unorthodox view
This question is similar to Overcoming Fear of Rejection, but not the same.
My experience in academia was teaching Computer Science for four years at a mostly liberal arts college, and that was 30 years ago.
I frankly found the world of work to provide more challenging intellectual stimulation, at least in the sort of practical issues I cared most about.
There are highly technical issues in software development that I and other practitioners have discovered that are quite orthogonal (to say the least) to the trend of publishing in those areas.
So when I try to publish it is hard to build much of a bibliography.
When I submit (through an online submittal process) I tend to get responses that vary in quality, but are generally negative.
I am tempted to revise, in such a way as to address the concerns raised by the better quality reviews.
I could buttress the claims with stronger mathematical arguments, but it's hard to tell if this would make the paper more or less approachable.
Then there's the question whether I should resubmit to the same journal.
There don't really seem to be any others at the same level of relevance to practitioners.
Sorry if this is a noobie question.

Being in CS myself, I find it difficult to give general advice without knowing the details. Maybe you could provide a little more detail for the CS-ers amongst us. One thing is that Software Development is not the same things as CS.

@Dave: I really wanted to keep it general, but here are some links: A, B, C, and linked posts. It's something that other practitioners have also found, but I seem to be the only one who's tried to publish it.

@Dave: You're right, it's only CS in the sense that CS departments are teaching something sort of automatically that I wish they would take a new look at, because their students are flooding the workplace with certain ideas I have a problem with.

@Charles: You asked "unorthodox way to solve a known problem, or did you find a problem that nobody else is considering as a problem". Well it's kind of both, I guess. The literature treats it as a problem having a pretty well-defined framework, for which all they can do is more deeply elaborate it, when in fact the practical form of the problem is qualitatively much different, needing a qualitatively different solution.

@Charles: The main criticisms (as I read them) is that I failed to make the case, and that being pointed out in detail tells me how to focus revisions. I did include a detailed use case, but it seemed disconnected from the argument. The other part of my question was if it's possible to re-submit, though they did not encourage me to do that. Your comment is helpful, and could be an answer.

@Charles: Another point (bear with me) is that it was suggested I do a survey of practical cases of performance tuning. I can't really do a survey. All I've got is generalities based on decades of experience and one concrete fleshed-out example posted on-line, plus rational and statistical argument. (It is a "Practice and Experience" journal.) So I'm a bit nonplussed about that kind of request.


text:
Do students need to wait a year to join university in Canada?
I am an Indian and my son is going to complete high school (+2) in March 2020.
I am planning to apply for engineering/computer science for my son in any university/college in Canada.
Now, I come to know that the application end date is January 2020 for the courses starting in September 2020.
In India, school final exams happen in March/April and results are published in May every year.
My son will get mark sheets, transfer certificate and all documents in May 2020, after completing his high school (+2) exams. When applying for a college/university in India (which opens in July), we need to submit above mentioned documents, because whether we are going to get the college seat or not is based on those documents/grade/marks.
So, do I have to wait to apply for him in Jan. 2021, for the courses starting in Sept. 2021?
In other words, do the students need to wait year to study in Canada?
As the schools in Canada ends in July/August, how they are applying to university/college?

Why are you applying? Shouldn't your son be applying himself?

Your question seems just target a specific people as indian student wants to join a college or university specifically in Ontario, CA. I'm not sure, this question could be answered here or not cause it depends on condition of those specific people.

@AloneProgrammer, I edited to make this questions as more common. How the local students handling this situation?

@NiceGuy Still it's not clear. What do you mean by saying: "I am wondering, as the schools in Canada ends in july/august, how they are applying to university/college?" You mean what documents are needed to apply or which site they use or something else? If you are looking for these kind of information I believe admission office of universities are the best resouces.

@AloneProgrammer, I added details in Edit part

@user2768, "I am indian" is relevant because start and end months for academic year is different in India and Canada. In India it is June-May and In Canada it is Sep-Aug.

@NiceGuy I think it pretty much depends on university's rules. Again, I reiterate my argument: The best bet for you is to contact admission office of the university that you want to apply to get help from them. Even different universities may have different rules...

@user2768: Why are you applying? :  This is the cultural difference, I hope no south-indians will ask this question. To clarify, We both will discuss, fill forms, apply together or spiliting tasks between us. We(south-Indians) don't see this approach as the son is incapable but an emotional attachment ( of we than I ) which made me phrase this question like that.

What does "high school (+2)" mean?

I edited your post to make it more readable. Instead of adding edits at the bottom, please edit the whole question as you go to incorporate necessary information. FYI, I'm not aware of any Canadian high schools that finish in July/August, but I could be wrong.

Canadian Students typically apply before they have received their diploma, and acceptance will be conditional on graduation. They send their notes transcript up to this point and any relevant documents. Please note that many programs have multiple admission rounds and applying during the summer is possible in that case. Good luck!

Based on other answers, comments and my own research, I found the below.
From University of Waterloo website...

Sending your documents
Once you've applied, there are a couple of
things you need to do before we can make an admission decision.
You must submit your transcripts and documents (outlined below) by our
document deadline.
You can upload unofficial documents so that we can review your
application and make a decision.
If you receive an offer of admission, we will require official versions of the documents. Your conditional offer of admission will provide details. We cannot make a decision if
we do not have all your documents, e.g., transcripts, English language
test scores (if required), Admission Information Form (if required for
your program). If your mid-term grades are not available by the
deadline, please submit them as soon as possible.

and

Students from India:
Please arrange for your Grade 10 board examination results and Grade
12 predicted examination results to be sent to the Office of the
Registrar.
If you receive an offer of admission to Waterloo, we will also require
your official Grade 12 board examination results.

Related Useful links:
https://uwaterloo.ca/future-students/admissions/sending-your-grades
https://www.ontariocolleges.ca/en/apply/important-dates
https://uwaterloo.ca/future-students/admissions/application-deadlines
https://uwaterloo.ca/future-students/admissions/official-documents
http://www.electronicinfo.ca/deadlines
https://www.ontario.ca/page/study-ontario-international-students#section-0
https://uwaterloo.ca/future-students/admissions/admission-requirements/computer-eng/international-system/indian-system/

The application deadline is January, yet your son won't receive grades until May. Perhaps your son can apply with his predicted grades, rather than his actual grades, and the university can make an offer conditional on him obtaining his predicted grades.

 I'm unfamiliar with the particularities of the Canadian system and my answer is based upon systems elsewhere.

Thanks for the reply, but I am looking for Canada specific info.

@NiceGuy Just check whether this applies in Canada.

I found more useful information here - https://uwaterloo.ca/future-students/admissions/admission-requirements/computer-eng/international-system/indian-system/

@NiceGuy - if your own research has given you a better answer than anybody else has posted so far; please don't hesitate to post your own findings as an answer (and accept it yourself). Doing so helps people in the future, who have the same problem as you.

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text:
Importance of research team in PhD
I have been accepted to two programs in education in the US. Both potential advisors are very active in research in different ways and advocated me a lot to the admission committee. The first uni has a very high reputation (top 50), and the advisor, an associate professor, is a leading expert in the field (Professor A). The second uni is top 160, and the advisor is an assistant professor (Professor B).
As far as I researched, most of Prof. A's PhD students are part-time students who hold other professional positions. In contrast, Prof. B's students are "traditional" and do the course in full-time mode. Furthermore, both profs often conduct research projects with other professors rather than students (evidence as recent years of research records). Therefore, I feel that I can have more chances of getting publications with seniors in the lab of Prof. B rather than in Prof. A's lab because A is busy and there are no full-time PhD students in the lab.
So my questions are:

If I choose Prof. A, what is your advice when working with a busy advisor? What should I do to get involved in more research projects and get publications during the PhD period?
If I choose Prof. B, to what extent does the ranking affect my career prospects (top 160)? To what extent can the publications replace the low rank? (How many publications, quality..)

It would be great to contact some students from  both teams and assess if your assumptions are really true. Publication records may not tell the entire story

I doubt that any program in Education actually functions as a lab. Much more likely to be individual work, I think.

@Buffy Education is pretty broad; on the side of ed psych and ed tech you'll see a lot more lab-like environment than ed policy, curriculum, etc.

There's a lot of questions here that are not entirely related. Please focus your Question on one question, although I'm not sure it will be on-topic.

Can you say more about the specific field within education? I might have a bit of general advice, but not for the question as stated.

For a warning about working with an untenured faculty member (assistant professor, say) see this question: https://academia.stackexchange.com/q/182213/75368

@AppliedAcademic I tried. However, as I mentioned, most of the students of prof A are professional so they are quite busy. I could only talk to one and they said that they don't do any research projects with the prof because they are busy with their full-time work. While students of prof B usually work on the projects with the advisor.

@AzorAhai-him- Sorry for not being clear. I edited the question.

@Hung Thanks for clarifying, but my multiple-questions comment still stands

@Buffy My area is Edu Policy

@AzorAhai-him- If I need to integrate to one question, this would be: How could I get more opportunities of involving in research projects (and have pubs) when working under a very busy advisor and a lot of part-time PhD students around?

In the world of research, publications matter more than university ranking. The prestige of the lab group is only good so far as it helps you in producing quality publications in quality journals. If you go to a highly ranked program with a fancy-pants PI but you fail to produce, no one will care about the former points. Go to the lab that you will thrive in and that will help you get the most quality pubs, is my advice. From what you have shared, it sounds like you already know this is group B.

+1 for go where you will thrive. Which professor does/supervises work you are more interested in?


text:
What should you consider before upload a revision to your arXiv paper when you spot an error?
I am in process of uploading a paper for arXiv. I have looked over the paper for several weeks now and it seems that there are no major issues, however, the paper is 24 pages long so I have accepted the possibility of having a few (non-trivial) typos and grammar mistakes here and there.
My question is, what should you consider before uploading a revision to your already uploaded paper, should you spot a mistake?
I am only asking because it seems that the people I have came across in my field rarely upload a revision, whether or not there are mistakes in their paper. One, for example, contained a glaring mistake in an important part of a proof, but the mistake is still there; I can see it now. The paper has been there for a year now, it has gone through a formal revision at a conference (where the mistake was corrected). But why not simply correct their arXiv copy?
Should I wait until more feedback is received before uploading a new revision? If so, how long? Is it okay to upload the revised version immediately or is there some arXiv mechanism that prevents me from doing so?
Is there a catch for uploading revisions?

Another reason is that the arXiv itself somewhat discourages overly frequent updates. For instance, after the 5th version(?), all future updates are no longer publicized in the mailings, so you might end up with the first few trivial corrections seeing more publicity than the future ones. I am myself no stranger to the "I'll update the arXiv once I've changed something significant" behavior, even if the preprint is years old and it is questionable whether any significant changes are to come.

Whatever.  Most people find arXiv papers through Google searches and citations, not through email announcements.  (I'm an arXiv moderator.)  It's far better to have a few less people hear about the correction than to leave the error in place.

Some journal policies can prevent people from uploading new versions of a paper based on feedback from their review process, at least for a certain time period.  Some journals in some fields will even consider an arxiv version a "prior publication" of the paper, and as such won't bother to even review it.
Even if there's only a time delay, it's entirely possible that the author(s) simply forgot, as after 6+ months they are probably swamped in new projects, new semesters, etc.  This can be compounded if the corrected version uses journal specific styles, and so would have to be edited to be suitable for the Arxiv.
Individual authors also use the Arxiv differently.  Some of them use it simply as a way to announce their results once they've advanced far enough for a preprint. Such people are unlikely to bother with updating, especially if the paper gets accepted by the first journal it is submitted to.  Others go further and view it as a paywall free way to maintain an accurate and up-to-date record of their work.  These people are likely to update with corrected and published versions whenever possible, and may view doing so as an important (or even necessary) service to the research community.  Many sit in between: they may update when significant changes in the results have been made, but won't bother with grammar and typo fixes alone.
In short: it's a matter of personal taste and preference.  You don't want to submit an update for each typo you find, but you should at some point have a sense of "meaningful amounts of change has occurred, and I do not think I'll be making any more changes in the near future", at which point it would be appropriate to update if so desired.

The first reason may well be non-existent. I have never heard of a journal halting the publication of a paper because an author uploaded a referee-corrected version onto the arXiv, let alone trying to go after the author in any other ways. (This is about "accepted manuscripts", not about the final versions formatted by the journal -- those even arXiv itself doesn't usually want.) This is not overly surprising, because the only people who could reasonably police these kinds of rules would be the editors, and those rarely have the incentives to harrass the research community they belong to.

@darijgrinberg Agreed. Such policies do exist, but it's dubious about how enforceable they will really be, for the reasons you state.  They are likely to discourage posting the update regardless, though, as the author similarly has no incentive to harass and antagonize the journals and editors they depend on.  They might not try to penalize you for that paper, but possibly they will be less receptive to future submissions.


text:
How do textbook authors and professors cook up homework/assignment questions?
I was looking around on the internet and I found some notes with homework appended to them (for those interested, it is math). What struck me is that these questions are not only lengthy, notationally challenging, but also appears to be quite complicated. For a biology course no less.
Can any authors or professors illuminate as to how (i.e. the process) these assignment questions are developed? I thought about this and found several scenarios which are quite interesting to think about, but at the same time, ridiculous to me.
For professors/authors:
scenario a: the professor/author goes home and quietly sits in front of a fireplace and starts brain storming weird scenarios and writes down these random thoughts
scenario b: the professor/author demands his graduate students to pump out 10 questions on a weekly basis, the rest is history
scenario c: the professor/author picks up another book and make adjustments to questions that looks interesting and then puts into his own book. But then this sort of becomes a chicken-egg problem. What about a completely new field that no one has touched before?
scenario d: there exist a large central depository of problems that professors submit to it, and draws question from it, a massive database out of reach of any students. This is some fringe conspiracy theory.
scenario e: The Hamilton approach. Ideas just flies out at random say when you are taking a shower, or buying some groceries and then you jog them down very quickly and then send it to a depository. Kind of like scenario a but much more random.
What about people who are cross discipline and working on their own field? Are there dedicated personnel who is neither an author, nor a professor, but just works on developing these assignments all day? This question just opens up a can of worms.
Please feed my curiosity, I would love to know how they are developed because maybe I will also develop my own someday. I know there exist some distinction between authors and professors (although the two coincides on a great degree), let me know if things done differently between different professions.

I've never been teaching a class but tutoring pupils from 10 to 18 years (mostly math and computer programming). When I create assignment questions for them I try to create question that makes them understand the concept they are currently studying and combine it with something they might find interesting. E.g. math text problems for a 10 year old might contain the name of their cat and a programming assignment for a 17 year old might be programming a command line game. I think most often one is influenced by already existing problems.

In the more complex case were the students are already university students there are also very often problems that are similar to already existing ones (you can't blame the profs, how else would you teach a certain algorithm like quick sort without having a programming assignment about it?) but the more a lecture/book is specialized the assignments/problems will be related to what the prof has seen in his research like @ff524 said.

I would assume all except (d) is true for most people at least sometimes.

Great question. If the answer is (d), I hope someone sends on the link.

Obviously it's varying combinations of these, depending on which professor and what sort of course it is.

Related: http://academia.stackexchange.com/q/41863/19607

While I am not aware of (d) on a global scale, it does happen on a smaller scale. If the same course is repeated over multiple years, but given by different persons, you can easily build up a database of questions that worked well.

@xLeitix: You should also exclude (b): it’s unavailable to many teachers, and some of the rest dislike it.

For scenario (c): I assume "picks up another book" is a misspelling of "does a Google search"

I have been teaching university courses, as well as authored books and booklets for students, in computer science. I have therefore created assignment questions for these books, exams, and term papers. I always found two qualities of an assignment to be the most important one: It must contribute to the understanding of the topic the textbook or term was about. Additionally, it must be fair, i.e., solvable in the allotted time frame without the student/reader having to be a genius. I usually create questions in this way:

I identify the essential topics for the textbook's chapter, or the term. These topics constitute the "general topic area" any question must reside in. I usually write these down on paper and put the back in front of me every time I need to create a new assignment.
For each topic:

I identify the most common concepts one is required to understand in order to master the topic. Basically, the building blocks of a topic.
I formulate a problem that requires the student to correctly apply this "building block" in order to solve the problem.

When I have created a set of questions for a a chapter of textbook, I save them and close the file for at least 2 days, sometimes a week. I create a usually calendar entry to look at them again.
When the calendar alarm fires, I open the file with the assignments and print it. Then I set a timer and begin answering my own questions. For textbooks, one needs the answers anyways, typically. I set a timer and, for each question answered, note how much time I needed for answering the question. I take care to only use the tools the student will also have at hand. I then multiply the time by 2.5 (roughly --- your mileage may vary, but see below). This gives me a rough idea on how long a student might need to answer my question.
I check the timespans against the allotted time for the student. Typically, there is some sort of time boundary for students: In an exam, obviously, but also for term papers and textbook chapters.
Refinement.

Note that the estimation of how much time your students may need to answer your question depends very much on how fast you answer your own questions, and is also a question of experience. I usually take my time and try to do them "extra-neat." Especially when creating textbook or lecture notes, this pays off quickly. This is how I arrived at the factor 2.5. As @scaahu pointed out in his comment, a higher factor might be sensible. In general, I'd recommend starting with a higher factor --- for example, the mentioned 5.0 --- and reduce it only if your students consistently have much time left after an exam, or are finishing their term papers very quickly, over a number of terms (this is important!). Forcing students to come up with answers quicker and quicker does not necessarily mean that your assignments will have made them smarter. More often, it creates a lot of frustration among the students without any benefits, and, in my experience, rather leads them to present (generic) answers they memorized beforehand from, e.g., your lecture notes or other textbooks, rather then come up with their own answers.

I like your answer except multiply the time by (roughly) 2.5.  Five(5.0) or even more is my own experience. Maybe your students are smarter?

This is partly experience. I have to admit that I usually take my time when answering my questions, e.g., I use drawing utensils instead of doing it freehand. Perhaps that accounts for some of the difference.

@scaaahu I've updated my answer to include your comment; you are right that my original "2.5" estimation can be misleading and lead to much frustration among the students due to them not having enough time for good/well-thought answers.

You missed scenario f, which I suspect is the most common:  The professor has been teaching this class and refining the textbook/notes for years, and developed homework questions out of problems that have come up over the course of that extended period of time of engaging with this material.

I also suspect this is the most common method, except I would have written "found" or even "stolen" instead of "developed".  (I have certainly "developed" a lot of homework problems over the last two decades.)

Actually, none of your scenarios describes my workflow. I use two approaches:

For every chapter/section/topic/… (let stick to section) I think about what concepts the students should master. To figure this out I collect all definitions, results and techniques from that section and design questions around these. For example, I think about questions that make the students understand a definition (and a definition is best understood while working with it). I am from mathematics, so here is a mathematical example: The concept are continuous functions. Problems for this concept could be "Show that this function is continuous!" with a concrete function or "Let f be some continuous function. Show that f has this or that property!". For results the approach to design problems would be to think about why you teach this result. It certainly is good for something. Find that and ask a question for you can use the result. For techniques it is even simpler: Give a problem that can be solved with this technique.
There are things that could be explained in the lectures but for some reason I prefer to have them in the homework. This could be a corollary/side result/additional result that is useful and simple enough to derive but too much for the lectures. It could also be a core result for which the proof is simple and the idea of proof is obvious or the proof is just very illuminating.

I should add that when I decided what kind of homework I want to give, I usually look at textbook and see if I find problems in the respective categories and only if I can't find problems that fit, I start to derive my own problems. (So you may say that after I decided which kind of problem I want, I follow scenario c…)

Another case is where you've worked on a more complex form of a problem in the past -- perhaps as part of a larger research project. Perhaps the answer had an interesting or elegant 'core' to it. You try to write a more simplified version of the same problem, while preserve the interesting essence of the more complex problem. Relevance to a current 'real world' problem helps to make assignment/tutorial problems more interesting, and they stimulate more thought and discussion.

This I believe is how my professors developed some of their homework problems.

To be honest, from my own experience and working with two different advisors of two different universities in two different countries (US and non-US), the professor will choose book A to include in the syllabus and choose book B (doesn't mention it to students) for homework and exams questions. For instance, for steel design, many professors like T. Segui's book to get examples and ask students to buy it because it is very simple and straightforward. However, most homework questions come from McCormac's book that is a bit more complicated. I have rarely seen anyone wrote their own exams except for very old professors who are "old school".

I've been amassing my own sort of "test bank" of question types and formula variations of my own design over the past several semesters, driven in large part by the desire to both create a pool of as many non-repeated question types as possible (to foil those amassing their own test files on the other side of the battlefield), as well as the interest in finding problem types that capture as much of the relevant core concepts as possible in a compact, relatively workable form (to reduce the number of students whose level of comprehension is buried under a tangled mess of bad algebra).
For example, as part of the calculus discussion on derivatives and their effects on the graph of a function, I've been slowly compiling a critical mass of equation types whose derivatives are reasonably straightforward (even if not always "easy") and which contain at least example of most of the key features (maxima, minima, inflection points, etc.), from which I can then pull my favorites (or favorites of the week) to craft a number of distinctive but reasonably equivalent versions of a test for a large section. In deference to the identify-key-details nature of my father-in-law's abiding hobby of birdwatching, I call it the "calculus field guide."  Work now to accelerate for the summer, no pun intended.


text:
H-index: Any quantitative (or other) measures on how well h-index actually measures scientific productivity?
I am wondering if there have been any sort of rigorous/meaningful study about how well an h-index measures the scientific productivity of a person.  I am wondering how much stock I should put into the h-index, both as a personal measure and a measure of those around me.

Having the ability to quantify how well the h-index measures scientific productivity would by proxy imply that there exists some reliable measure of scientific productivity to use as a control. Unfortunately, no such reliable measure exists.

Is the h-index intended to indicate productivity at all? I think a researcher can be tremendously productive (in terms of publication frequency, number of findings, etc.) and yet receive very few citations for one reason or another.

Relevant ( see the answers for why not to do this) : http://academia.stackexchange.com/questions/5687/how-to-find-the-h-index-of-a-professor/5705#5705

Why on earth would you want to measure scientific productivity anyway, as opposed to scientific impact, or better yet, scientific insight?

@JeffE, depending on how you define scientific productivity it could include those aspects as well.  At least that's what I've been thinking, maybe I'm wrong and that's what's producing some of the confusion here.

Besides the convincing argument in Moriarty's comment to the OP there is also a formal argument that the index is particularly meaningless. The Notices of the AMS recently published "Critique of Hirsch's Citation Index: A Combinatorial Fermi Problem" by Alexander Yong, see here http://www.ams.org/notices/201409/rnoti-p1040.pdf
Roughly: The h-index is about half the square root of the number of citations...
More precisely: For an author with N citations to his paper, the h-Index always lies between zero and the square root of N. Now assume that these citation are distributed on a number of papers and all partitions occur with equal probability. Then it turns out that the estimated h-Index is then about 0.54 times the square root of N. In the article it is quoted that Hirsch himself has observed that basically all h-Indices he looked at were in the range of 0.45 and 0.58 times the square root of N. Yong also states that, especially for the highly cited people in math, physics, computer science and statistics only very few people have an h-Index above 0.54 times square root of N (and they are never more that 5% above that). The article contains much more data but only for famous mathematicians such as Field's medalists or members of the National Academy of Science and it is really worth a read.

"Roughly: The h index is about the square root of the number of publications..." Is this math-specific? Because for CS that seems awfully low.

Oops, should have been "number of citations".  Corrected. But then this applies universal - check the paper, it is really worth a read!

Shouldn't it be that the h-index is roughly half of the square root of the total citations? @xLeitix One of the early and easy-to-understand discussions is here: http://michaelnielsen.org/blog/why-the-h-index-is-virtually-no-use/ It was empirically observed in physics (not math) by Hirsch himself when he proposed the index. The AMS article gives more insight into it through combinatorics and observes that (a more elaborated version of) the rule of thumb works rather well for pure math. I think it should work more or less alright for other fields if citation patters are similar.

[ (citations)^(1/2) ] / 2 indeed seems a reasonable approximation for the h-index for most researchers I checked. For really large h-indices (> 40 or 50) the rule seems to get iffy, though.

@xLeitix In fact for higher number of citation the estimates tends to be too optimistic, i.e. the h-Index is usually smaller than that; also observed by Hirsch and documented in Yong's paper.

@xLeitix By definition, you can compute a scientist's h-index from the total number of citations and how much each paper contributes to the total. Now, assume that a given discipline has a typical pattern regarding the distribution of each paper's contribution to the total citation number. Then, you should be able to estimate the h-index of a typical scientist in this discipline from the total number of citations alone. In other words, if you know the general pattern of how a person's papers get cited, surely you have an estimation that takes the total citations as the sole input.

The AMS article used a simple random model for the citation distribution, and gave a simple one-variable estimate (i.e., about 0.54 of the square root of the total citation count). And it turns out It doesn't seem to be far off in pure math for the cases they checked. This estimate seems to work ok for physics as well, which might suggest that how physicists get cited isn't that different than in math. In any case, if there is a typical citation pattern in a field and if you model it well, it's no wonder someone's h-index in your field typically follows this kind of one-variable estimate.

@YuichiroFujiwara I think the interesting thing is that 1. the possible maximal derivation is not that large and 2. the probabilty of larger derivations is extremely small in theory and also in practice. Also interesing: People with more citations tend to be more below the estimate.

@Dirk I totally agree. What intrigued me more though is that the simple random model works so well. This could just be a coincidence, though. I couldn't care less about h-index and whatnot as quality/quantity measures, but how people read/cite/write papers and how networks evolve are interesting...

I'm no expert on this, and this is just a comment that is a little too long to cram in the comment section.
Before addressing your question, h-index is not supposed to measure productivity alone. It attempts to measure both productivity and impact in terms of citation count. The simplest and crude measure of the former would be the number of papers a scientist has published, and one simple measure of the latter would be the total number of citations. You could say these are two crude measures of quality and quantity, although their meanings are vague. In a sense, h-index is intended to measure the quality and quantity of scientific output by a single number.
Anyway, to answer your question, assuming that you are asking if there's any serious research on how well h-index works as intended, if you just google with some obvious keywords, you should be able to find many serious academic papers that study h-index, e.g.,
A. F. J. van Raan, Comparison of the Hirsch-index with standard bibliometric indicators and with peer judgment for 147 chemistry research groups, Scientometrics 67 (2006) 491-502.
or several papers cited in this article (pages 5-6)
or many other papers you should be able to find through usual means like checking the references at the end of a paper on the topic, looking up papers that cite them, putting more effort in googling, trying different keywords you couldn't come up with at first but you can after reading some papers on the topic, using different search engines, etc.
Note that the above parers are by no means meant to be representative examples. I just googled a bit, skimmed bits and pieces, and copy-pasted the links. As I said at the beginning of this post, I'm no expert and as clueless as you. For what it is worth, the journal Scientometrics seems to have a decent impact factor, since you seem to care about that sort of thing.
I hope real academics who work on bibliometrics chime in and give expert answers.


text:
Making future goals clear in statement of purpose when changing fields
I need your advice, dear visitors. Currently I am finishing my BS in Applied Mathematics and Computer Science, and trying to apply for a PhD program in Computer Science to universities in the USA.
But I have a real problem.
The topics on which I worked with my adviser are purely mathematical (differential geometry). I began to study this branch of math 2 years ago. But 1 year ago, due to a suggestion of my friend, I started reading articles on Theoretical Computer Science (namely algorithms on planar graphs and in Linear Algebra but nothing related to numerical methods) and became carried away by that. For some reasons, I couldn't have abandoned the work on my topics. One of them is that the results in differential geometry we eventually obtained are excellent (together with my advisor we have two articles in good mathematical journals). Since this is the last year of my education at my University, I haven't changed the field for my diploma (it would have had no sense and it would have been impossible in some way, but let me not tell about it).
Differential Geometry is very interesting and I know much about it but I feel that algorithms is the field to which my future research will be dedicated. I am not interested in any other field of CS. By the way, my scientific advisor knows about my future plans (reported to him half a year ago). He also heartily agreed to write a strong letter of recommendation for me.
So what have I got? Good mathematical skills: a wide spectrum of knowledge on many fields including algebra (linear and abstract), advanced analysis, differential geometry (as mentioned before), functional analysis, combinatorics, algorithms, graph theory... I also have medals and prizes earned on mathematical olympiads (IMO, IMC). So math and TCS background is not a problem for me. After all, Theoretical Computer Science is the same math (we also state and prove theorems like in math)!
Why a PHD? Because I absolutely know that one-two years spent on getting a MS finished will be just a waste of my time.
So I feel difficulty when composing a Statement of Purpose. I just don't know in which way I should put the description of my personal experience forward in such a way so it would be clear and laconic. I don't want the reader to be confused about my goals. I don't want my statement of purpose to be a fruit-salad. I want the reader to understand that a PHD in CS is what I need since I am completely determined about my intentions.
A lot of sincere thanks to whoever will answer my question. Maybe, one can give me a link to a sample of SOP similar to my case.

Have you asked your adviser?

I think you've essentially written an outline of your SoP in the question. Namely,
"I'm interested in theoretical computer science and mathematics. I've worked in differential geometry, and here are my results, but I'm now fascinated by algorithms research, and I have extensive math background".
Now just expand on it.

¶1. I have an extensive math background.  [Details]  ¶2. In particular, I have published results in  differential geometry. [Details]  ¶3. I am fascinated by algorithms.  [Details]  ¶4. My math background is relevant to my algorithmic research interests. [Details]  ¶5. I want a PhD in computer science because... [Details]  ¶6. I want that PhD from your department because... [Details] Done.

Two pieces of further advice: (1) Do not write "I am not interested in any other field of CS." in your statement of purpose; that is a serious red flag. (2) Apply to my department!

Darn it. I forgot to tell you to apply to my department as well :)

The best advice any of us could give you is to do an internet search for what others have written, and use such examples to help structure your own personal statement.
Essentially, you want the following points:

What previous experience (particularly research) do you have in this field?
What are your future research goals? How do these goals tie in with past experiences?
Why do you want to do this research?
Why should you be chosen above other applicants?

No one here is going to help you write it or proof-read your statements. If you have further general (i.e. not field-specific) questions that are well structured, they are much more likely to elicit a good response than the floundering "where do I start?" that I see here.


text:
How do I deal with graduate school rejection from an university that I considered easy to get into?
Things you should know:

It was not a dream school.
It was considered "safe/moderate" i.e. the kind of students who get in
have more or less similar profiles like mine.
From graduate school student discussion forums , seems like no one else got
rejected.
This really hurts. If this school rejects me, what hope do I have with others?

You should analyse your application; if the factors that are easy to compare like grades, subjects, etc. leave you wondering then maybe your cover letter or your letters of recommendation were not compelling enough?

@T.F. Could it be because I specified a very specific interest area which I thought the grad school was into. Is it possible that they rejected because it doesn't fit into what they do?

Do you care to share the degree you sought after and possibly the school. Perhaps the studies were highly competitive fields and the application was submitted late in the review process and the spots were filled?

Did you score well on the GRE? How was your undergrad GPA? There are many many factors...perhaps it is no "one thing" My suggestion is to simply do your best to try again.

@Phlume Yes my GRE was well above average and my undergrad  CGPA well was not outstanding..but then this university was judged as a good bet for my profile!

seems like no one else got rejected — Rest assured, either many many others were rejected, or you should be happy they let you go.

Speaking from purely personal experience, I wouldn't necessarily worry about it. Of course it's possible that your application was weak and that generally the other schools will feel the same way. On the other hand, it's also possible that something you wrote just didn't appeal to someone working on the applications at this particular school and you were looked over. In my own experience applying to graduate school I actually got rejected from some of my 'backup' schools but was accepted (and was given extra incentive to come to) one of my 'dream' schools.

It can all depend on the mood of the person(s) on the day they happened to review your application.

Graduate admissions are very different than undergraduate admissions. In undergraduate admissions, you can generally predict where you'll be accepted and where you won't. But graduate admissions - particularly PhD programs - are significantly more complicated. Because of quotas and financial obligations, they have to try to pick, on the first round, students that they actually think will attend. The reason for this is because, once they make a financial offer, they know you can hold on to it until April 15 or something, and during that time, they can't allocate that money for anyone else.
So, if they see someone who is easily qualified for the school, but they suspect will get other, better offers, they'll likely just pass on that person in the first place, since they don't want to commit money to someone who won't end up attending. If they are mistaken about this, they may end up with a small and/or weak incoming cohort.

(I would have merely posted this as a comment in response to gammapoint but my lowly reputation will not permit me.)
I was accepted by every program I applied to except my safety school.
My experience when applying to undergrad schools was pretty much the opposite: I only got in to one non-safety school.
My guess as to why something like this might happen is that sample sizes are a lot smaller (fewer applicants to a given grad program in a given year than to the undergraduate school), so your results are more likely to differ from what you expect.


text:
For those who study in the humanities, have I made a mistake in doing my Master's abroad?
I'm an American student currently earning a Master's in American cultural studies in Germany. Why study American culture in Germany? Firstly, I wouldn't have been able to afford a U.S. Master's in this field, and secondly I'd like to learn/improve my German with the ultimate goal of (possibly) pursuing a PhD in comparative literature.
However, since classes have started here, I'm beginning to regret my decision for the following reasons:
A) classes are incredibly large, direct contact with professors is scant, and so, among the obvious academic disadvantages, I'm ultimately worried I won't be able to get good letters of rec.
B) even at the graduate level, many students don't take their studies seriously (probably because there isn't the pressure of student loans and because education is quite open here [a good thing])
C) classes are not as rigorous as they were in my B.A. literature program, and so I don't feel like I can grow as a researcher/scholar, which is my goal
Does anyone have a similar experience applying to American PhD programs after doing a Master's abroad? Should I just stick it out and try my best to befriend some professors and make the best of my time here?
At this point, I regret not applying directly to American PhD programs sans Master's degree since my undergraduate grades and professors were quite good.
Thanks in advance!

You may have made a mistake, but it isn't a fatal one. You need a path forward.
First, your undergraduate professors can still have credence in writing recommendations, but you need to keep in contact with them so you aren't forgotten. They can probably give you advice now on your current situation.
I assume that you are early in your European studies. Abandoning and starting over in a US doctoral program might be an option, though perhaps as last resort.
The B) issue can, perhaps, help you with the A) issue. Professors might be a bit tired of lazy students and could possibly appreciate an approach from a student who wants to do more, both helping with the letters of recommendation issue and your C) problem.
But, it won't work if you are passive. You need to take charge. If one of your current professors does something interesting, you could make an approach and offer to help in some way. This solves all the problems. But don't just "stick it out". Take charge.

A) ... I won't be able to get good letters of rec

Are you required to write a dissertation or conduct a large project or similar? If so, then the supervisor of that work will be well positioned to write a letter of recommendation.

B) ... many students don't take their studies seriously

So what?

probably because there isn't the pressure of student loans

Isn't there? How do students fund their studies?

because education is quite open here

What does that mean?

C) classes are not as rigorous as they were in my B.A. literature program

Was your previous university more prestigious? That may explain (B) and (C).

I'll write a thesis, yes, but graduate programs require three strong letters of recommendation, and my thesis advisors would only provide one or maybe two. As far as how they fund their studies, German higher education is quite cheap and has robust financial aid. Some may have loans, but the majority do not go into debt to fund their education.  I pay, for example, a little more than 300 per semester, which includes a state-wide public transportation ticket. When I say that programs are more open, I mean that they are less selective and larger.