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Is is possible to study mathematics PHD with another bachelor degree like Business Administration?
Is it possible to enrol in a mathematics PhD program with a bachelor degree in Business Administration? Have you heard of someone doing something similar and if yes, how did they do it?
I am a BBA student. I found that I love mathematics after taking some analysis courses and want to keep on studying mathematics and doing research. However, since I didn't really enjoy studying BBA, I have a poor a GPA and I suspect I won't be able to enrol in a competitive mathematics PhD program.
I currently have some background in analysis, including metric space topology, some basic differential manifold, functional analysis, real analysis and of course some other basic math courses.
What is the best way to continue my academic career in mathematics coming from an unrelated field?

Have you considered preparing for/ applying for a master's degree in mathematics first? I suspect PhD programs might be a bit leery, but you could "prove yourself" in a master's program, and gain some more sophistication in math (though it sounds like you're off to a good start with what you've already taken).

I believe Ed Witten was a history major as an undergraduate. Preparation is more important than your major. If you do well on the GREs and have a solid foundation in undergraduate mathematics you should do fine.

@amWhy I wonder if the university will admit someone with low gpa to their master programme.

Low GPA can be explained. A low math GPA is a lot harder to explain. The mathematics subject GRE (if you're in the States) and a good recommendation, preferably from a well-respected professor, are key.

@amWhy Is it possible for a non-math major to get into a masters pogram in US?

I'm starting to think we need a reference question "Is it possible to get into grad school in X with an undergraduate degree in Y?"  See these other similar questions.

In my answer I had the impression you had already graduated, but now I see you mention you are a student. Where are you in your study - junior, etc? It might not be too late to take enough courses to get a math minor if you don't want to switch majors, but that's a very program-dependent option and every school is unique in what they accept/require. Also, MANY programs automatically ignore the first 1-2 years of grades and just want to know about your last 2 years.

The short answer is: "yes, but it won't automatically be easy"
First, every department of every school in every major has a set of preferences - things they like to see in an applicant. This list includes GPA, GRE, sometimes subject-specific GRE, specific coursework, specific reference letters, and the rarely-ever-listed undergraduate research/term papers that lots of people here give great advice about but that I didn't previously know was a 'thing'.
Looking over the admittance pages of prospective programs can give you an idea of this at times. In your case, I looked at a few schools math departments I'm familiar with, and using this one as an example because its not terribly atypical (but you should research your own preferred institutions): University of Wisconsin Mathematics

1) A GPA of at least 3.0 overall and substantially higher in mathematics courses.
2) 18 or more credits of mathematics beyond calculus. The 18 credits should include at least one semester of both rigorous analysis
  (sometimes called advanced calculus) and abstract algebra, and
  preferably two semesters. Advanced undergraduate courses in other
  topics further strengthen your application.
6) A “Supplemental Application Form” is required after submitting your
  online Graduate School application. It is important that you include
  all advanced mathematics courses you have completed, are currently
  enrolled in, or intend to take...
7) Three letters that address the question of mathematical promise.
  These letters should be from mathematics faculty, or other faculty
  familiar with the demands of graduate work in mathematics...

Now keep in mind that any published outlines are a "wishlist", and they don't automatically toss anyone. But every area you don't fit requires explanation or some other counter-balancing factor. Some programs give a list of x requirements, then say they'll waive 1-2 of them without further explanation needed, etc.
If you are missing all of these, then you need to look at a different program, but you might find no program that's a fit. If this happens, that doesn't mean you can't do something, just that you might have to go about it sideways or backtrack a bit.
If you are missing just a few things, like a course or two, some programs have probation where they'll tentatively admit you on the demand you complete X courses on your first semester and get at least Y GPA, etc.
Alternatively, you might need to make arrangements to take advanced math courses at an undergrad/graduate level. It's harder to get financial aid for this sort of thing (you can't qualify for a Pell grant anymore, etc), but you can still get loans or pay for it outright. If you need this route, you need to talk personally with professors in advance to discuss what your goals are and make any possible arrangement to do extra work, write an academic research paper, etc. You need to find people you can have enough contact with to be willing to go to bat for you in writing strong reference letters if you don't already have math professors who are familiar with your past work. Join math clubs, go to math study labs, volunteer, whatever you can manage.
Before you go running off and doing any of this, however, you should make contact with math faculty at your present school and/or at a target school, and get an opportunity to talk with them about what you want to do and get their advice. Sometimes just showing "mathematical promise" can get a magic wand waived over your application, which maybe you already have - but only someone familiar with what you've done can tell you that. If you have a piece of mathematical "best work" from your most advanced class, bring it with you to such a meeting.
Once you've gotten a few honest appraisals of where you stand, be kind to yourself in looking for programs that will fit YOU, not some imaginary ideal program for someone else.
Finally, one last piece of advice. Keep your mind open to hybrid possibilities. There are whole fields of study you might really like that you aren't aware of, including applied mathematics study offered by business schools, economic fields that emphasize mathematical theory and/or application, etc.
Sometimes we have to try to mold ourselves to fit an existing mold a little better, but sometimes its trying to fit a square peg into a round hole. Talk to people and investigate the possibilities, get a clearer understanding of your relative shape, and seek out openings with geometry congruent to yours. Good luck in your journey!


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Publishing in one vs. many journals
I have started publishing papers this year - 3 so far, of which 2 are in an international journal, the 3rd in a local journal.
My question is, is there any particular disadvantage with publishing the majority of papers in one or a very limited array of journals? Or is it a case that it makes no difference either way?

Getting published in a good journal is what is most important. Impact factor is usually considered the most important way to evaluate a journal. However recently publishing is undergoing change with the proliferation of Open Source. Who knows how much impact these journals will have in the future?

I am sure opinions and traditions vary concerning this question. For me the quality of the journal (say, impact factor) is the most important, you try to get published in good journals. In some cases, the field may be so narrow that only one or a very few journals may be good. In such cases the selection may result in a single journal.
To publish in a variety of journals seems to be a means by itself for many but I think the reasons still vary. There may be advantages in getting published in many journals to show that the research has wider applications and is accepted more generally. It may be a way to avoid the suspicion that you have a back way into a single journal (not that that would be true). In some cases, you may select different journals because you know your ideas are not favoured by someone in a specific journal's editorial board (strategic reason).
So reasons for spreading the publications may vary quite substantially. The main point for me is, however, still that primarily the quality of the journal will decide. I would not chose a different journal just to get a new journal mentioned in my list of publications.

That is a great point of taking the quality of the journal over the quantity of journals.  That is something that is very good to know for up and coming researchers.

In my field (mathematics), the quality of journals is a quite fuzzy notion; we do not rely very much on metrics like the impact factor, but have a subjective and qualitative sense of prestige of journals. Inevitably, the way a given journal is regarded changes from one person to another. As a consequence, publishing in a variety of journals increases the odds that someone looking at your publication list will think "whao, she published in X!".
Another point is that if one publishes a large fraction of its research in one journal, people can wonder whether she has a friend in the editorial board that help her getting accepted there.
All in all, it seems preferable (and possible, given the large amount of journal in the field) to publish in a variety of journal for hiring, promotion and more generally evaluation matters. Of course, the picture is certainly different in other fields.

I agree with @PeterJansson that impact factor is certainly an important factor in selecting a journal. But I would also think about who you want to reach with your paper. Although Google Scholar and similar search engines now ensure a much higher probability of finding your paper based on its title, abstract and keywords, it will only be found by people actively searching for terms associated with your work.
Most researchers still subscribe to specific journals in their field, even if only through e-mail alerts, so I target a journal also for its audience (while trying to select the highest impact factor from this subset). This is especially important if your work bridges several fields.
For example, my research is about human-robot interaction, so I have to decide whether to publish an article in a journal that is read primarily by roboticists or social scientists/psychologists.
In this case, it is also vitally important to adjust the paper to its target demographic. Psychologists don't know (and likely don't care) about my implementation details while roboticists are more likely to be interested in nitty gritty details about the code and less about the nuances of the social psychology theory behind it.

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Should I mention my reference's university role (administrative) when listing?
I have a list of references in my CV. One of them happens to be the Dean of the graduate school and the other a Vice Chancellor. Both of these roles are administrative and their primary position is that of a Professor. Would listing these referees as
John H. Doeberg
Professor and Dean of the Graduate School
University of Foo

Jane Q. Publicowski
Professor and Vice Chancellor of Academic Affairs
University of Bar

be better (give more credence/weight) than simply listing them as a Professor. I know that ultimately what they write in those letters is what matters, but for the purposes of this question, it can be safely assumed that both letters are going to be stellar recommendations.

The thing to remember here is that your reference letter writers will be providing official letters, and therefore will be including their full titles (presumably the letterhead will list this). In such a case, it doesn't really seem to make much sense to "hide" the status of the person writing the letter. Moreover, if the position is one which will include some administrative duties, a letter of reference from someone doing university administration might even be a plus.

You haven't said what the letters are for. If it's for a position based on academic credentials (whether in academia or industry), I doubt it makes much of a difference. If the position involves administrative responsibilities, then mentioning this would help, so the people requesting the letter would make sure to ask them.

They are for an academic/research position with possibly minor administrative responsibilities...

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What does "First Class or 2.1 Honours degree" mean?
A PhD position for which I am planning to apply notes:

Applicants should have (or expect to achieve) a minimum of a First Class or 2.1 Honours degree.

The position is in the UK, but I live in another EU country. What does "First Class or 2.1 Honours degree" mean, and how do I know if I satisfy that condition?
The grades at my university are given as fail, 3, 4 and 5.

The "definitions," such as they exist, of the different degrees, are available from Wikipedia.
Essentially, "first-class honours" and "2.1 (or 2:1) honours" are the top two categories of honors degrees in the UK system—sort of like the summa cum laude and magna cum laude in American universities. Although the requirements for awarding them appear to be fairly uniform between schools (with the possible exception of Oxford and Cambridge—see dbmag9's comment below), given that each university may have different standards for how they "translate" degrees from other countries, your best bet is to contact the university's international office (or equivalent), to determine if they will consider your degree to be comparable.

I wouldn't say that the requirements for a First or 2.1 'differ substantially' within the UK; it's pretty consistently 70% for a First and 60% for a 2.1. What might differ is the difficulty of achieving that normalised percentage, and how much work will have been involved in the degree generally.

Don't be confused by the low percentages mentioned by dbmag9: results are typically normalised lower in the UK than many other places. Typically around 10–12% of our undergraduates manage a 1st, and around 30–40% a 2:1.

I had to achieve >= 90% for a 1st


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Etiquette of publishing (continued) research after graduation with an MS degree?
Situation
I am soon (maybe...) going to graduate with a thesis-based MS in a technical field. The past year and a half has been incredibly frustrating from a research perspective, as my advisor has insisted on telling me exactly what and how to research, essentially prohibiting me from exploring any of my ideas which are not approved by him.
I believe this is because I unfortunately chose a research area for my MS directly related to his PhD work from some time ago.
I have, for purposes of finishing my degree program, simply acquiesced to his requests in the hopes of finishing as this degree has dragged on for far longer than I want and my current goal is to simply be finished with school officially.
I am currently working full-time (after working in research on campus for ~2.5 years) and have no intent to pursue a PhD in the near future and if so, absolutely not with my current advisor.
Question
I do have personal interest in my research and can see myself exploring my own interests post-graduation, which could result in publishable material. I have no interest in giving my current advisor credit for any of this, should it happen. I also understand he likely would be pretty upset should I actually publish related to my MS work without his involvement.
Are there standard etiquette, unspoken rules, or things to be aware of, within either the academic world or research world which address this situation?
I am posting this anonymously but will be able to clarify

Did your advisor make a significant intellectual contribution to your work?  If so, you cannot publish that work without him.  If not, it's your work; do whatever you like.

@JeffE to my thesis work, I suppose. But to any research which I would be doing post-masters, no.

The big issue here is the following: will you need to have your master's thesis advisor write you a letter of recommendation in the near future? If so, then it will be potentially a big problem to publish material from your thesis without his involvement. This will provoke a response, which could be damaging to, for instance, job applications or graduate school admissions. So tread carefully before trying to publish the MS work.
However, since your advisor has proscribed work in other directions as part of your thesis, you could certainly explore those on your own. In such a case—since it isn't an outgrowth of your thesis work, but essentially a "parallel" work, you wouldn't need to share co-author credit with him. This is especially true if you write the paper and direct the research yourself; in such cases, he has no real claim to being on the paper.

This answer presupposes that advisors are automatically coauthors, which is not true in all fields.

@JeffE: Not really: the only presupposition here is that the advisor is a control freak. Even if you're in a discipline where the student could publish independently, if the advisor is left out of the loop but finds out about it, I think the student will still be in a bad situation. Not necessarily in every situation, but definitely in this one.


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Can I have both my supervisors write reference letters?
It seems like most PhD applications require at least two reference letter. One of my Master's thesis supervisors (a PhD) has written me a reference letter. Can I ask my other supervisor (a PhD student) to write me the second one, or would that be redundant?
The other alternative letter writers are all professors who have only seen me in class and can't really comment on my research potential.

I've sat on my department's graduate studies committee during application season, and so I can provide at least one point of anecdata for how PhD application recommendation letters are used, at least in my tiny corner of the academy. YMMV.
First, These letters are among the last things that are looked at. We don't have rigorous thresholds for grades, GRE scores, subject GRE scores, publications, etc. But all of those things are used to "objectively" sort the pool. First the obvious admits are skimmed off the top, and fellowship offers are made to some of them. This is the first place where letters are important, since quotes are mined from those letters to try and win funding for these students from the University (i.e., we entice the student with a prestigious award, and the department doesn't even have to pay for her!) The money quote is usually a ranking and/or direct comparison -- "In my twenty years at Giant State University, I've seen no more than 5 students of X's caliber. I would say her talents remind me of Y's at that age, who is now a professor at Prestigious U" -- and this is why even a mediocre teaching letter is more useful for this purpose than a nice letter from a PhD student (who can not be counted on to have a reasonable intuition for another's future potential).
Second, we work our way down to the marginal cases; this is much more difficult since the marginal cases require a more holistic take on the application package. No one gets immediately thrown out just for bad grades or a bad subject GRE score, but we expect some explanation of a mitigating circumstance (not an excuse!) in the personal statement, and hopefully reflected in the letters as well. For example, if you were dinged in the first pass for a low GPA, we would like to see that it was a bad first year (and not a bad fifth...); that you are on an upswing. If you think you might be a marginal case for whatever reason, the letter from the PhD student might be helpful -- perhaps you don't have any publications, but the PhD student can vouch for your work and verify that you have something in preparation, or explain why your project failed (projects often fail!) but that nonetheless you mastered some state-of-the-art technique on the way.
On the other hand it is a big waving red flag if those things are then not pointed out by your actual boss (the PhD holding supervisor). If you think the student will write a better letter, then you're probably best off having that student lobby on your behalf with the professor, to make sure the professor's letter is strong, and then get a second letter from someone else.
I would strongly recommend you do not ask for a letter from a non-academic unless it was an actual research position (and even then, these can be iffy if the supervisor has no letter-writing experience). On occasion we see glowing letters from a student's supervisor at their summer construction job, or a call center for a political campaign. Well-rounded students are great, but such letters are almost useless.

I wouldn't restrict things to "actual research positions." For example, in engineering, I would give a letter from a professional engineer substantial weight. In fact, some of the best letters I've seen have come from non-academics. (Writing good letters, furthermore, is not the exclusive province of academics! And some academics can't write very good letters, either.)

These letters are among the last things that are looked at. — Then your admissions procedures are very different from mine.  We don't admit anyone without strong letters.

@JeffE:  I understood that sentence as meaning that other factors are first used as a filter, rather than that they actually admit anyone without looking at their letters.

@TaraB: There's a big difference between "among the last" and "not the first".

Fair enough, @JeffE; although we should be careful with what we mean by "strong letters." I've certainly seen students accepted who had steller credentials, but I thought the letters were "weak" in the sense that the writer had no idea how to write a reference (or the student made the mistake of asking a famous person who has never actually interacted with them). We certainly would be wary of students, regardless of credentials, if the letters were negative in their content. This is not what I typically mean by a "weak letter".

Also it might help to clarify that I'm not at a "top" program, so I have no idea if this is a cultural difference (I'm not in CS), or a "taking what we can get" heuristic that has been developed.

The main point of the letters of recommendation are to comment on your abilities, and to place them in the context of other people the recommender has known and worked with. The better they can comment, and the larger the number of people the recommender has worked with and therefore can compare you to, the better the recommendation will likely be.
For this reason, it is not nearly as helpful to get a letter from a PhD student as from someone who is more experienced and has likely worked with more master's students. On the other hand, if the option is to get a letter who can only comment on your class work, I would opt for including the PhD student's recommendation letter.
However, if you have had a non-academic person (perhaps an internship supervisor or similar) in a related discipline who could comment on your work, this might be a suitable alternative. (See wsc's answer below.)

As it was mentioned before, you can obtain reference letters from your supervisors, in case that you have done some internship job for obtaining your master´s.
Also you can talk directly with one of your past professors, and explain them about your research interests (try to check first which professors have the same research interests). Then you can ask, in a nice way, if they can give you a hand with a recommendation letter. Most of the time they will agree to help you with that.


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Do media interviews enhance or potentially spoil academic reputations
During my research, asides from getting papers published, I have been interviewed for the radio and newspapers. My question is, are too many media 'appearances' a potential hindrance to someone early in their academic career?  Or is it also a case of 'all publicity is good'?
My main concern is being misquoted, which thankfully has not happened, though they exaggerated one (minor) point a tad.

The simple answer is that it depends on in what context the appearance occurs.
If you become visible as an expert in your field your appearances will build your reputation among the public as an expert. As such you will gain a positive (trustworthy) reputation which I am sure will be seen as positive in many respects. The university will gain credit by having employees appear as experts In my country all faculty also are mandated to convey science to the general public and appearances in media is one way to do this. So my guess is that the department and the university as well s your own discipline will see this as beneficial.
Obviously, if you appear with politically incorrect views it will all be negative. Another problem that I am sure will appear is jealousy from colleagues who do not appear in media. In the end your success depends on how you get portrayed in the media. My university runs media training to ensure that faculty represent the university in an appropriate way.
In the end positive media visibility is not likely to strongly affect your research career in terms increased funding, the systems should not really be influenced by it but we all al human. Media visibility will mainly affect the way you, your discipline, the university etc. is perceived by the public. It may, for example, influence the way politicians distribute research funding.
So, media interviews can potentially both enhance or spoil a career depending on the context in which you appear. I believe it is important to know how media works and universities often have programs (and sometimes also rules) to help in such contacts. Depending on the type of question to which you are requested to respond you need to be able to assess how answers will be used or perceived. If the interviews are on non-controversial topics, it is probably not problematic but if you are approached on a burning issue, be aware and be savy

Excellent points - it is sometimes a bit of a minefield when it comes to these kind of things. It is a case of stcking to the topic and the facts of the topic.

In the end positive media visibilty is not likely to strongly affect your research career — I disagree.  The PhD students you work with have a very strong effect on your research career, and publicly visible researchers attract significantly more PhD applicants.


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White Paper & MS in Mechanical Engineering?
I am a Mechanical Engineering student & want to pursue Master of Science with specialization in Design/CAD. Recently I have come across white papers on website of some companies about their products. My friend told me that publishing a white paper would be beneficial for my MS applications. So I wanted to know:
1. What exactly is the scope of a white paper?
2. What sort of topic/information must one put in the paper?
3. How to publish a white paper (I mean is there a Journal for white papers or any other standard)?
4. How much effect does it have in showing my research aptitude compared to a scientific publication/ research paper?

In the research field a white paper is not so important. However, if you plan to work in an applied field,a white paper is valued highly because it shows you are aware of the current issues in your field. In general, white papers are issue-oriented.
For example, in the field of architecture "green" building materials are valued over traditional. But why? A white paper could present facts on "green" building materials versus traditional materials in terms of cost and performance. Other factors could also be weighed such as customer preference. Therefore, when the executives of a firm read the white paper it could help them make a major decision on materials they will buy in the future.
See below the definition of white paper:

White papers in business-to-business (B2B) marketing
Since the early 1990s, the term white paper has been applied to documents used as B2B marketing or sales tools. Far more commercial white papers are now produced for B2B vendors than political white papers for governments[citation needed].
B2B white papers are long-form content designed to promote the products or services from a specific company. As a marketing tool, these papers use selected facts and logical arguments to build a case favorable to the company sponsoring the document. B2B white papers are often used to generate sales leads, establish thought leadership, make a business case, or inform and persuade prospective customers, channel partners, journalists, analysts, or investors.
Many B2B white papers argue that one particular technology, product or method is superior for solving a specific business problem. They may also present research findings, list a set of questions or tips about a certain business issue, or highlight a particular product or service from a vendor.

Please do not sign your posts, especially not with a link to your website included; every post you make is already linked to your account. Self-promotion is frowned upon on the Stack Exchange network. Just stick to the relevant material in each post, and put a link to your website in your profile page if you want.


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progressing from high school teaching into academia
I have been a high school teacher for coming on 14 years, in 8 schools, in 2 countries. While teaching still is enjoyable, I feel with the near completion of my PhD (in physics), that I am ready for a new challenge in academia.
I have no illusions though, it will be challenging to even get into academia.  I know I also have to be published (have 3 peer reviewed papers published).
How should the skills developed in high school teaching be touted to be relevant for academic postions?

Are you wanting to focus on teaching, research, or a combination of the two?

Ideally, research, but I'd be happy with a combination

If you are near completing your PhD, then I assume you also have some papers coming out soon. If not, then your dissertation will probably be worth 3 papers, so I wouldn't worry too much about that part.
Conveying information to an audience with varied skills and backgrounds is always a valuable thing in academia. Whether in written manuscripts or conference presentations, this is usually the problem to be solved. My advisor was fond of constructing lectures, no matter the audience, as if they were to be given to college freshmen. I'd say a lot of freshmen are not too different from high school students.
A successful PhD defense signifies that you have the ability to do novel work in your field, but your teaching skills will enable you to tell others about it.

Indeed, that is what I was thinking, but was dissuaded from applying for a position recently, despite the qualifications needed being an Honours or a Masters (have both), because of their view of the distinction between high school and freshman.
I am very fortunate that I have aleady got 3 peer reviewed papers published.

+1 for a lot of freshmen are not too different from high school students


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Motivating students to push their limits
I think I must be stranger than usual sometimes, I always seem to take on topics that are somewhat more difficult than the mainstream, in very trying conditions - but the outcomes are very much worth it (not for me per se, but the benefits for everyone, potentially).
I try to instill this in my students, by 'gently' mentoring them and encouraging them to push their own limits a little further each time.  Also, I teach them that 'failure' is just another step to success.  For the most part, my students take on the challenges and the work, ideas and enthusiasm from them, frankly humbles me.
However, I am always wishing to learn new techniques to encourage my students to push, and even exceed their limits.
What strategies are most effective for encouraging students (and colleagues for that matter) to take on the difficult topics?

As a reflection on my own experience as a teacher, I am convinced that motivation is not a static, unchangeable property of particular students, but it is a multifaceted concept, a variable state of mind, created through the interaction of the student with the subject matter in a particular environment (teacher, group, topic, etc). A good teacher arouses the motivation indirectly, by creating the right environment for learning. The students' effort ensues almost magically.
See this interesting article:
Linnenbrink, Elizabeth A., and Paul R. Pintrich. "Motivation as an enabler for academic success." School Psychology Review 31.3 (2002): 313-327.

This answer might be 100% accurate, but it somehow didn't give any useful suggestions for how to achieve that goal.  "A good teacher arouses the motivation indirectly" - OK, how?  What techniques work, and what don't?

The most important aspect is the teacher's attitude. It sounds as though your students are already benefiting from yours.
As @Cinco says above,

motivation is not a static, unchangeable property of particular students... A good teacher arouses the motivation indirectly, by creating the right environment for learning.

Based on my experiences with some awesome professors, I will say DO

hold your students to high academic standards.
assume that the students are capable of doing more than they are currently doing.
show them how the subject you are teaching is relevant to them.
be enthusiastic.
recognize their great work, and always expect more and better in the future.

Perhaps most importantly, have fun! Enjoyment is contagious, and a good professor can make it fun to learn about anything. A great professor can make it fun to struggle and push against the limits of our abilities.
(Sorry, this is rather subjective...it is hard to quantify what makes a great teacher!)


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Is a combined BS/MS in math a good idea for pursuit of PhD?
If I want to continue mathematics into a PhD program, would doing a bachelor's and master's combined program be a good idea? Would I be required to retake some graduate courses in the PhD program? This would add only one year to my degree from my current uni as opposed to the two from doing a BS and MS separately.
A bit of background information: I am a dual enrollment US student who has completed two years worth of college courses, and so I'll be entering as a freshman with junior standing taking upper division undergrad math courses. Thus, a standalone BS would take me only two years, leaving me to graduate at age 20. Is this too young to start applying to grad school?
Also, I plan to do an undergrad thesis; is one year of familiarizing myself with the faculty too little to choose an adviser? I'm not even sure which math I would want to write about. Will three semesters of intro analysis and intro algebra grad courses be enough to decide on the thesis for my fourth semester?

Mathematics is HUGE. You cannot call yourself a mathematician if you have not mastered (i) complex analysis and (ii) abstract algebra. Before then, there is no point in trying to decide what to do next, as most other topics will have bits and pieces of these two. If all you have had is calculus (without proofs, in the worst US route mem tradition), then you have a long road ahead of you. (Disclaimer: I had my abstract algebra and complex analysis in high school in Russia, which was enough to impress my measure theory teacher in my Ph.D. program by reproducing a proof from my 10th grade class.)

These classes are required in my curriculum, so I will fortunately be taking these soon. Do you consider complex analysis a prerequisite for real analysis, or vice versa, or a corequisite? BTW, I've been studying more advanced topics on my own since summer break started because I already know that calculus 1-3 is nothing compared to higher level mathematics.

Real analysis first seems like a more natural procession, but this may depend on how deeply your profs will go into their particular versions of these two major topics. Ask around -- come to the office hours of the profs offering these courses; profs LOVE curious minds (I used to :) )... especially given how few American students are really interested in math.

Well that's good for my schedule since I plan to take real analysis first. And I will make sure to get involved with the professors after class hours, considering they've already dealt with what I'm going through! Thanks for the advice.

@StasK Why complex analysis and abstract algebra but not real analysis?

@Jack, depends on how you define it -- I keep forgetting that say French theorems about extrema of differentiable functions are a part of real analysis in the U.S., not of calculus. (I had them as a part of a proper calculus sequence, Russian way: notion of the real line - sequences - infinite sums - single variable calculus with epsilons and deltas that relied on real analysis and sequences in say proving Bolzano-Weierstrass. American sequence, Stewart-style, makes little sense to me.) But my main point was about other areas having bits and pieces of these foundation topics in other areas.

@StasK Ah, that explains it. salute to the Russians. Had no idea it was like that elsewhere. Seems pretty tough for first/second years maybe?

Tii, what do you mean by combined program? What is wrong with a regular 4 year bachelor's and then 2 year master's?

@Tii, BTW, I started my Ph.D. at the age of 27, so there goes the quality of my advice :). I had two Master degrees by then though.

I finished my undergraduate degree at 20, and I know a number of others who have finished undergraduate considerably younger than I did. So, no, 20/21 is definitely not too young to apply for and begin a graduate degree program.
Since you list yourself as a US college student, here are a few questions to ask yourself before embarking on a dual-degree program:

Who is going to pay for the MS part of the program? You may not want to have to pay out of pocket the extra costs of the master's, when it would normally be covered if you picked it up "on the way" in the PhD program.
Are you happy with your choices for a research advisor for your master's thesis? (No point in being miserable by working for the wrong advisor—even if it is a semester or two, a miserable time is not what you want.
Where do graduates of your school's master's program end up for PhD programs? Are these schools you'd want to go to? (You can ask the same question of the bachelor's program, of course.)
Who will write your letters of recommendation? Do you have people who know your work beyond the classroom well enough to recommend you for external graduate programs? If not, this may vote in favor of staying for the master's program.
Do you have to make a decision about which program to do before you apply for external programs? No point in making a tough choice if you can keep it as a "fallback" option.

As for topics and coursework, I will leave those answers to the mathematicians.

Thanks for the input! I do hope to get to know higher level math better so I can decide what my thesis, undergrad or graduate, should be on, and choose my adviser according to their research. Also, do you know if there's a site for checking where people go from undergrad to get their PhD? I'd like to see where students from my college went to, but it seems that that information is not present on the school website.

I don't know of a website that does this—however, your department's undergraduate advisor may have this information.

Alright, I'll go ask him about this-- he did mention having the statistics of all the students! Thanks again!

@Tii, there are people I know who get bachelor's at 18 or 19. Just saying.


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Academic bullies
A question if I may, about appropriate methods of dealing with academic bullies (or as one of my students call them "wannabe nemesis").  What would be the best, most dignified way to 'overcome' them, while protecting your own research interests andreputation?
To clarify what I mean by an academic bully - first and foremost, I am not referring to those that offer critiques in any constructive forms (even in an aggressive manner), I am referring to the academics that who, for whatever reason (usually due to disgreeing with research findings or feeling threatened by 'newcomers' to the research field), decide to go on the offensive with insults, threats, and the like.
Finally, I am referring to, if the communication is direct (by email. letter or in person).

As you point out there is usually some personal (psychological) reason for such behaviour, particularly if it is repeated/continuous. I believe the first thing to do is to avoid to go by insinct and "fight back". Obviously the correct response will rely on the type of communication/accusation but keeping a calm and cool tone is the best. Keep all replies very short and avoid emotional touches at all costs. "Thank your for your opinion." (or something along those lines) is enough to have provided a response while practically ending the correspondence since there is nothing more to reply to. To try to "kill" the conversation by not providing fuel is the way to go. In verbal communications a simple "OK" can be enough. Then try to walk away at the first opportunity.
Bad behaviour is otherwise best fought with good science and academic professionality and one must try to find moral support in situations like this so that ones self-confidence is not shattered. There are many destructive behaviours that affect us, some temporary some part of (or deficiency in) a personality. There is however, no reason to go into mud-slinging with such persons unless they express clearly unethical opinions but then the question is at a different level.

it also helps as well, when you are able to get support from colleagues and/or organisations.


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What could be a possible next step after a bachelors in Economics and Business Administration?
I've degrees in International Economics and Business Administration.
I consider myself an entrepreneur, certainly my past degree did give me perspective but I feel it lacked solid practical knowledge, so in my entrepreneurial career it didn't help me much.
I was thinking to go for computer science degree. But having bachelors degree in Economics/BA it will be hard for me to go directly for masters? (even though I've fairly good knowledge of the subject). Restarting bachelors degree in a completely different field could be a total waste of time for 26 years old.
Also I'm aware that allot of students do it the other way around, first they get science degree and than they do masters in business administration. Do you think my way will be harder and more intense? as master in computer science can be much challenging than masters in business administration or computer science bachelors?

Where are you from btw? Would you be interested to move somewhere else or are you limited to your area?

@T.F. I'm from Republic of Georgia, I studied business administration in Hungary and Economics in Italy. I also speak four languages, so my preference is to move around :)

I can think of a couple of options off the top of my head:

Apply for a Masters in CS anyway and see how you go, you might get credit for your experience.
perhaps do a postgraduate certificate/diploma in CS (if available) and use that as a stepping stone to the Masters degree that you desire.
Perhaps consider a MBA that has a strong focus on CS (as a major).

It would also depend if you are intending to pursue your proposed Masters by coursework, research or as a mix of the two.

Regarding point 1, I can hardly ever imagine someone getting admitted to a CS master with zero CS classes (besides some math probably) in undergraduate studies. I like option 3 though, there seems to be a wide variety of MBA programs available.

So very true on point 3, as your post below demonstrates.

Another option is going for a masters degree in business informatics, a lot of european universities would admit you in it. I do not know if this kind of discipline is big where you live, but it is one of the biggest disciplines in a lot of universities in germany and austria. A study from the university of vienna has shown that graduands from business informatics earn the most in the industry and have the least problems to get jobs out of all graduands.
You would have to take roughly one semester worth of courses in algorithms, software development etc. extra, but that should neither be an intellectual problem, since you already know a lot about it, nor a practical problem, because you will get a more academic and thus systematic idea of what matters in computer science.
Even if you got into a computer science master right away, well how would that help you? You seem to have many practical skills from CS, but you would likely end up doing it just for the degree, because the students who have a bachelors in CS will most likely be much better prepared for doing the necessary methods for a masters program in that field. If you want the pure CS, you are likely much better off just starting with a bachelor, maybe aside working if you are already really good at CS.
If you care about how hard a master will be, well that can't be answered right away. But you have to keep in mind, that economics is a social science and thus not as exact as a technical science.

The last sentence is interesting. Is business informatics social science or technical science?

Multidisciplinary I would say, but in your masters you can often choose between courses and could end up having a profile that is very technical.

This is very interesting. And in fact I considered this as an option. I wasn't able to find good university with with CS oriented MBI, I looked into switzerland unives particularly in Lausanne - EPFL seemed like a good spot, but their line-up of English only courses is very limited and it will take me some time to learn French.

Is states best for this kind of Education?

As a serious question: what do you hope to gain from the additional degree?
You haven't really indicated why you want to get a master's degree in computer science, other than "it's the next step." You also haven't indicated why you think you'd be able to get into a master's degree program (do you have enough CS courses to convince a committee?).
But, most importantly, the question to ask yourself is whether or not there is anything really to gain from the two years or so spent earning a master's degree that you can't obtain via another route. If you can do that, then it's probably worth trying. If, on the other hand, it's just to improve programming skills, there are lots of better ways to do that than to spend two years in CS (which doesn't really do much in that regard).

Well, one good reason I've is that local degree makes it easy to move to the other country and later find a job.

+1 I think this is the big question. If you just want a degree in a country, then an MBA with a focus on IT would be the easiest to enter (and complete).


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How to carry a scientific poster as hand luggage?
Sometimes I have to present at different conferences. Usually I print the poster at the print office at home and cary the poster with me to the conference. With the more and strict luggage rules, it is not always allowed to carry the usual chunky poster cylinders on board.
As I like to travel light, this sometimes is a problem. One solution would be to look for a print shop at the destination (eg. Paris or Berlin).
But then again, you also want to bring back the poster with you.
So my question, how can I bring a a1/a2 poster with me as carry-on luggage?


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How much code to include in a Physics paper?
This question is related to Physics research papers, particularly one that will be coming up for my ongoing PhD research (4th paper in all).
A core part of the research is developing, testing and validating relevant self-written code.  So, my question is how much of the code should be included in the upcoming paper?  Should it be the whole thing, or a written summary of the main components/subroutines used?

Did you ask your advisor ?

Yes, and we're both wondering, as we've seen papers that include some code, some that included all code and some that had details of the main subroutines used.

I think that if you are discussing the code or using it to illustrate a point, it can go into the paper, but preferably only snippets. If not, it should be in a repository somewhere where people can download it, eg github bitbucket or similar. If you put a lot of code in the paper it adds clutter, making the paper harder to read, and is also harder for people to run the code. I mean, cutting and pasting code from a document is no fun.

You might also try asking on scicomp.SE.

Regardless of the content of the paper, it would be great if you uploaded your whole working code to GitHub (with some full examples of usage).

If the goal of your paper is to show that your code solves a particular problem or performs a specific task, then it is your responsibility to prove to the reviewers that your code does what it claims to do. The easiest way to do this is to include the code as part of the supporting information for your paper.
However, reproducibility—which is what you're asking about, to a certain extent—is a not-so-simple question when it comes to showing codes do what they're supposed to. For instance, a code that runs one way under one configuration may return a slightly (or perhaps _very) different result when run under another configuration. This doesn't mean that one result or the other is wrong—it just means that this behavior needs to be taken into account when evaluating the correctness of software.
One way to help this is to provide as much information as possible on your testing environment, so that any differences between the system on wich you were working and the conditions the reviewers and future potential users of your code will have can quickly be identified. This would be included, perhaps, as a text file that accompanies the code in the supporting information.

very good points, particularly on the testing environments - which would not only be the 'external' environment (field testing etc), but also the 'internal' environment - each subroutine of the program itself.

As long as there is input and output should be easy to debug the code. Of course, THERE ARE EXCEPTIONS...

@NPcompleteUser: The issue here is that there might not be a bug in the code, but simply how the code was compiled and executed could make a substantial difference. In my field, for instance, you will likely get very different outputs if you run on different numbers of processors—although the long-term averages you get should still be consistent. So you can't simply say "my run does not agree with yours, therefore the code doesn't work!"

I would second @NPcompleteUser's answer about having the code and some sort of walkthrough for reproducing your results online. However your question about how much to include in the paper is very journal specific. There are a number of publications specifically for computational science with formats where large chunks of code are expected, but if it's a journal whose focus is more on the science than the techniques, I would just describe in words the tricky parts of your algorithm and give a footnote reference to the website where the code lives.

I would put the current version [of the code] on gist. Providing inputs and gnuplot (matplotlib/whatever) scripts for graph reproduction.

Would that also work if the inputs are photos/pixel data?

as long as it is tar-gzipped - should be fine. Usually I also describe each file in the archive, so that GH stuff doesn't have to guess if I break any copyrights.

Papers on physics which utilise code should primarily describe the algorithms and their relevance to the physical problem at hand, and not focus on nuts and bolts of the code. There are several reasons for this: the printed article is the wrong place to include code, as page space is generally limited and not suited to formatting of code; there are issues with copyright as in most cases the ownership of the code is transferred to the journal, or if open access then put in the public domain; generally the audience of a paper should be broad enough to be useful to a wider range of researchers than your specialist area.
As others have suggested, you should release your code on GitHub or similar and apply a suitable licence, and simply refer to it in the paper. There are some exceptions to including code in a paper, generally when understanding the algorithm is essential for the reader or complicated, in which case pseudo code is acceptable since the essential information is included without any language specific distractions.

Another option, in the field of astrophysics, is to submit your code to a repository, and register it in the Astrophysics Source Code Library (ASCL) at https://ascl.net, which is indexed by the SAO/NASA's Astrophysics Database System (ADS), the official compilation of literature (and now source code and datasets) in astrophysics.
If that's your case, you look at this link to find out how to submit your code to it: https://ascl.net/submissions
The page has additional resources telling you how to cite said codes, and this is what a citation might look like:

Garrido, J. et al., 2013, AstroTaverna, Astrophysics Source Code Library, record ascl:1307.007

With the record being held at https://ascl.net/1307.007, and the ADS entry at http://adsabs.harvard.edu/abs/2013ascl.soft07007G


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Overcoming nerves from anticipating conference questions
My field is atmospheric physics.
The irony is that I have been a school teacher for over a decade, but soon, I'll be giving a presentation of some of my findings at a conference.  I think that the nerves stem from speaking about my own research in front of my peers - something that I have not done to a large audience.
The questions that will no doubt be ask fill me with anticipation in both positive and negative ways.
Asides from being prepared, making sure the presentation is seamless and that I have a good night's sleep beforehand and 'knowing my stuff' inside and out.  What are some strategies to anticipate the type of questions that would likely arise from a conference presentation?

Have rehearsal in front of your colleagues would help.

Write down questions that you think the audience may ask, and have answers or backup slides ready for them.

There are a few obvious questions to ask yourself in planning for questions:

What are the inherent weaknesses in the current work? (Almost no research is completely "airtight," so figuring out where the weak spots are will make a difference.)
What are the ramifications of whatever assumptions I have made? Are they logical? What happens if I strengthen, relax, or eliminate some of those assumptions? Will everything still work in the more general (or more restricted) case?
How would I apply this work to other problems? How will it help others in the field?

And then, with respect to the presentation:

Have I left anything out in the interests of time that would potentially interest the user? Is the research methodology clear?

If there's anything in the last point, you may want to plan on having additional "backup" slides which highlight that info, but that aren't part of the "main" talk.

Thank you for this, (+1).  I totally agree that there is no such thing as an 'air tight' research, and the additional backup slides would logically have information pertaining to your first 3 points.

One strategy to adopt when answering questions is to first repeat back the essence of the question to the questioner:

If I understand correctly, you are asking ....

This will have two effects. Firstly, it ensures that you are actually answering the correct question. Secondly, it will buy you a little bit of time to gather your thoughts and think a bit about an answer.
Take your time when answering questions, rather than rushing to the first answer that pops into your head. In the end, it is okay to say, "I don't know" or to ask to discuss the question off-line, but the latter can seem like a bit of a cop-out. Try to answer the question, but only if the message is not getting through can you suggest to take it off-line.
But the real key is to practice your presentation extensively. If you deliver a good presentation and you know it, you will feel great and thus comfortable to answer questions.

Similar in effect to a well prepared and executed lesson.

really good advice

On a side note, repeating the question also makes sure the rest of your audience knows what the question was. If someone in the front asks a question (without a microphone), the people in the back will most likely not have heard it.

In addition to strategies for anticipating questions, I thought it would be helpful to suggest how to cope with the nerves. I find that it sometimes helps to remind yourself a couple of things:

Remember that you have been working and thinking about your specific question probably more than most people hearing the talk - they are just hearing about your work for the first time. Even if there are important and smart people in the audience, you are the expert on your work.
Personally, I find that I am similarly anxious before giving talks regarding my work. However, it somehow always plays out fine - the atmosphere is usually relaxed and the questions tend to be either simple clarifications or interesting discussions. I suspect I am not the only person who experiences this.

Yes, in the smaller scale speeches I have given, this tended to happen.  I switched into 'teacher mode' (my day to day job)

Most of the questions you'll receive will either 1) ask for clarification about your methodology/results, or 2) suggest avenues of future work. Questions of the first kind are usually very easy (presumably you did the research and know the answer ;) ) and the second kind can be very helpful for identifying new research directions and collaborators! If you've already thought about the proposed direction, obviously chip in any insights you might have, but if not, "that's an interesting suggestion, and something I'd be interested in looking at in future work" is all you really need to say.
There are only a few realistic ways the Q&A session can go off of the rails. You might get questions like

"How does your work compare to [Foo et al 2003]" (you have no idea who Foo is or what his method does)
"Does your work account for (some factor you don't understand)?"
"How might your work apply to (some area you know nothing about)?"

You can fall into the trap of feeling that you should know the answer, and that admitting ignorance is embarrassing... but the worst thing you can do is to bluff or make stuff up. If people "smell blood in the water" and get the impression that you are being misleading, they will come back with even more hostile questions. Instead, remember that you are in control of the conversation, and hold the ultimate trump:

"Unfortunately I'm not familiar offhand with the method of Foo et al, but I'd be happy to chat with you offline about it after the session."
"I don't know offhand, but I'd be happy to chat more offline."
"Unfortunately I'm not too familiar with (area X), but I'd love to talk to you later about possible applications of my work there."

There is absolutely nothing wrong with not knowing the answer to a difficult question; if we know the answers to all the questions, then by definition it isn't research.  Being comfortable with not knowing the answer should help with nerves.  The questions are generally asked out of genuine interest, rather than as a test, so the person asking the question is not necessarily expecting you to have a good answer anyway. User168715's suggestion (+1) of saying "I don't know offhand, but I'd be happy to chat more off-line" is a good one", and is a good way of exchanging ideas with others interested in the same sorts of work as yourself.


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Funding with PHD +MS admission
I have completed my BSc in physics  and now doing my MS in my country(Bangladesh), but I want to do PHD/PHD+MS in USA with decent funding (RA or TA).
I know In a PHD level, funding may be available, but what about MS?
I'm saying this because I want to apply in this fall-2014 and during application I may not get my MS results .

If you're already in an MS program now, you're probably not even eligible to apply to an MS program in the US, at least in the same field.  Just go for the PhD.

But I have not finished  the course yet and during the application, I will not be able to show my MS results. Will they accept me in  a PHD program without MS?

Yes, of course.  Most people apply to degree N+1 while they are still working on degree N.

And, at least in the sciences in the US, earning the MS is not required, and many do not do it. I applied to a PhD program while I was finishing my BS. I never earned the MS. Don't worry that your MS is not complete! Just go for the PhD!

In most cases, you do not have to be a MS graduate in order to apply for a PhD. I was a MS candidate when I applied for PhD about 4 years ago. I could only show 2 semesters worth of results from my MS program. It works.
Also, in most programs around the HCI/information science space (and if I might venture to propose, also in computer science), there is very limited funding for MS only programs. Things might be different in physics. I will leave it to the folks who know physics better to comment.

In the US, you can apply to PhD programs without having a master's degree; this is actually the case in many fields. However, you will have to successfully complete the requirements for candidacy to a PhD program before being admitted to the doctoral phase of the program.
Funding for PhD-level programs at reputable departments (at least in science and engineering) should normally be guaranteed for some fixed term, provided you are making adequate progress and satisfy all program requirements. (What you have to do for such funding—whether it be research or TA—may vary, but that the funding will come from somewhere should be stated in advance.)

Whether a US university would accept you into a PhD program would depend on the University itself - you'll need to contact them and ask them directly.
Having said that, do you have semester results in your MS, in your application you could state these and say that the final results are pending.  Ask the universities you apply to if it is alright to submit your final transcript a little bit later.
In the meantime, gather all your academic credentials that may help in your application - published papers, conference presentations (if any), references etc


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Secure backups of PhD and journal-research
Common sense dictates that any and all work/research/drafts/etc that we do in our journal-research and PhD ought to be backed up and backed up again.  Most of the time, this is constrained to laptop/desktop, USB memory and perhaps a copy in your own email - one disaster could wipe all that out (almost happened to my MSc research - got caught up in a severe natural disaster).
My question is, is there a more secure and 'safe' means to save/backup our PhD and/or journal-research work? Do universities offer such services, or is it up to individual researchers to find their own solutions?
I should note that I have no illusions of any system being completely 100% safe and secure, but are looking at ways that are more secure than the 'traditional' methods listed before in the first paragraph (2nd sentence) of this question.

I'm not sure to see if it's very relevant to academia, and not just a backup problem. In that case, it might be a better fit for http://superuser.com.

@CharlesMorisset surely a question regarding tools that would allow for secure robust backups of data, drafts and he like would be relevant for Academia?

I am also referring to the discussion in meta here http://meta.academia.stackexchange.com/questions/443/are-questions-about-tools-on-topic

What are the "traditional" backup methods "listed before"? If the reference is to "laptop/desktop, USB memory and perhaps a copy in your own email", then the defails of said backups are not clear.

@FaheemMitha clarified - see the addition to the last sentence in the last paragraph.

Perhaps not the answer you were looking for, but publishing your work is in some sense the "ultimate backup". This is why I try to put every relevant piece of research into the manuscript (much as supplemental material) and submit as much data as possible to public databases. That way you only need to backup what you are currently working on. Of course this also helps people who want to further explore my published research.

@Bitwise that's actually an excellent answer, I agree, it is the ultimate backup.

@Damien: We have had some discussions in the past about "boat programming" questions: http://meta.academia.stackexchange.com/q/459/102. I'm not saying your question is not relevant, I'm just wondering whether you wouldn't get better and more accurate answers at another SE, since it does not seem to be particularly specific to Academia.

@CharlesMorisset I have never herd of 'boat-programming' and are still a bit confused after reading that link.

Further detail can be found here, but the idea is to decide if a question is very specific to academics, or if it could apply to many other categories. For instance, to exaggerate, you could ask "what is the best car for academics?", using the argument that some academics do use a car to go to work. Such a question would clearly be off-topic. In your case, it's not clear whether it's such a situation or not, and that's why I didn't close the question, but merely suggested that it might not be very specific.

@CharlesMorisset - I still don't quite get the boat-programming thing or why this question is a screw-up or whatever.  This question was intended to find out the strategies employed by other academics.  It seems you are not the only one who has concerns with this (2 downvotes have occurred).

@CharlesMorisset have clarified by making explicit references to PhD and journal-research

Not an answer for you, but my lab have automated daily backups of our lab computers. I would expect more experimental labs, using lot of big data, to have such services, but it seems to be rare.

@BenoîtKloeckner definitely a good example of where backups are absolutely critical!  I agree, it seems to be disturbingly rare.

@BenoîtKloeckner: good point. We don't have automated backup of the lab and office computers, but the file server which we anyways use to transfer original data from the lab (instrument) computers to office or computation server has automated back-up (and I also rsync my projects there for backup). I've met that configuration also in other places.

Backups, short-term and long-term storage of research material is actually something we have to address whenever we apply for a IRB for a research study or an IRB exemption. We have to spell everything out in great detail including when we are going to destroy or get rid of the acquired research materials (since they usually contain personal identifiers like name, address, email etc.)
This is our approach:

We have our own encrypted lab server on a RAID configuration. Access to the server is restricted to myself and my adviser.
We store generalized, public versions of our scripts on github. The previous answer has very good specifics on github so I will not go into the details here.
Periodically, we backup our lab server to another externally hosted server (not physically located in the US). This backup script is encrypted and password protected.
Physical interview questionnaires, survey instruments, interview transcripts are stored in a locked file cabinet with access to my adviser and myself in our lab. Entry to our lab is through keycard only.
Interview recordings, videos and audios are also stored and backed-up according to the procedures in 1 and [3].

This approach seems to have worked so far.

Having only two people knowing the password makes the Bus factor very high (plus the more secure it is, it is easier to loose access to it). So there is often a trade off (surely, for some confidential data it may be better to loose access to it than to disseminate).

Sure, but we don't develop commercial software. We write scripts which implement various algorithms which we derive mathematically. Said scripts are uploaded on github which everyone can see. Our collected data is protected.

For most cases, using git or hg is too much for research that don' t include much code.
For a concise answer:
Put everything in a web service like dropbox that provides function of version control. Dropbox is easy and straightforward to use and don' t bother you with complex version control commands, while it provides file version system for last 30 days. The free space is usually enough for current work.
For code, you can put at bitbucket.org, since it offers nice unlimited plan for academic use, which offers you private repos and unlimited team member. Of course you can open source your code at github.

Note that with a cloud service (except mega...), you should expect the NSA and other agencies to have have access to your research. This is usually not a problem, but in some cases that can be a relevant issue.

In the case with NSA concerns, you can setup your own file syncing server inside your lab using some open source solution like seafile.

In my institute it would be a concern (even without NSA). So we use an internal git server with access rights for collaborators. IMHO git is also good for research that includes writing text, not only source code.

AAAAAAAARGH be careful of sync services like Dropbox that don't version! The horror scenario (and it's happened to people!) is that you fumblefingeredly delete something, Dropbox syncs the deletion, and then the file exists in NEITHER PLACE.

@D.Salo I believe Dropbox keeps a copy you can restore for some time. As long as you realize you accidentally deleted something, you should be alright.

This is not really a research-related question at all, but I can tell you what I do. I put everything (excluding large data sets) under distributed version control (Mercurial) and push to remote repositories frequently.
NB: You can easily automate the pushing if you desire; this has positive and negative aspects.
Of course, the two leading DVCS are Git and Mercurial, so either of these would work. Free hosting for Git is available at Github and Bitbucket, free hosting for Mercurial is available on Bitbucket. There are many other places you can put your repositories, of course, including remote servers your organization owns.
Note: large data sets require special handling, so I'm not sure about that, but they are often not original work, and can be retrieved from other places if necessary.
I strongly recommend doing this in conjunction with some sane method for saving "scratch" work (i.e. work in progress that is not yet suitable for a fully formed commit), so you don't have significant unsaved work lying on your hard disk. The important point here is to have a workflow by which you can sanely push commits at frequent intervals - I try to make commits every 1/2 hr or less. I use Mercurial queues. There is also a brand new thing called the Evolve Extension, but I have not tried
that yet. Git has similar software available. See for example, the Stack Overflow question git equivalent to hg mq? and What's the Git approach to publish a patch queue?. NB: If you do use a patch queue, you need of course to version it, otherwise you can't push it.
These methods can appear complex if you are not used to version control. However, I can attest this approach works well. My workstation died three times in the last 3 1/2 years, and each time I was able to switch to a different machine and carry on working without interruption, without access to the machine I had just been using.
See also Why use version control systems for writing a paper, particularly user244795's answer.

To add a slightly less "public" view (my institute is a non-university research institute which would be counted either as academic or industrial depending on whom you ask). We mainly work with original data we measure ourselves. The major concern of our administration seems to be that original data does not inadvertently end up becoming public, destroyed/lost data seems to be less of a concern (but there are backed up servers, see below).

Default policy is that data and code (and paper drafts) stay inside the institute (though we may take them home, as working from home office is a valid and frequently used option) and within the collaborating groups, respectively.
Basically this has not only the obvious consequence of no google docs, no public github or the like (unless specified for the particular project) but also e.g. no private github plan.
Instead of private github plan (or the like) we have our own git server. The main purpose of the server is not backup (it is located in the main server room, so e.g. a fire killing the file server would most likely kill the git server as well), but collaboration. However, having collaboration with a DCVS automatically means backups that are distributed as far away as the collaborators are.
OTOH, our computer services provides a file server which is automatically backed up.
Once a publication is out, original data, source code and the paper are backed up in a publication data base as well.
Due to home office, I have an external hard disk that I carry hence and forth (particularly for data that is too big to be conveniently exchanged via internet with the git server). Which means that unless the big disaster happens when the external hard disk, my laptop, my desktop, and the file server with the backups are all destroyed (what is the probability that I'm well and up to work after such an event?), there will be some copies surviving.
At a university where  I worked before, the central computer administration did backup services for specified computers of the institutes. We once needed that after a fire in our institute's server room.
I may add that from my perspective of a scientist who is also doing measurements on physical (in fact, biological) samples, I've experienced much more trouble due to physically destroyed or spoiled samples than with computer failures.
Sure, we have our share of instrument computers dying (big issue, as it can be difficult to find parts that are compatible particularly with older instrument hardware - but no data loss issue: measurements are usually immediately transferred to file server, office computers, external hard disk). Office computers seldom die without warning symptoms beforehand. There also was that fire in the server room (university computation center back-up did work).
In the same time, we've had 3 failures of deep-freezers where our tissue samples are stored. Plus other accidents (tissues thawed by inexperienced student, lots of tissues never frozen correctly by collaborating surgeons, ...).
So even without a "proper" data backup plan like automatic backup of all computers, physical loss of delicate samples seems to be the more pressing problem than loss of data.

The classic digital-preservationist mnemonic for this is "3-2-1."

THREE copies of everything, on at least
TWO different types of storage medium, with
ONE offsite.

The storage-medium thing is so that you don't get bitten by (e.g.) a bad batch of one manufacturer's hard drives (cough cough Seagate) or a general weakness in a particular medium (such as USB drives' propensity to get lost or break). Offsite is so that a building fire or earthquake or tornado doesn't do you in.
The lowest-stress way to handle this is typically a backup-designated hard drive in your office with automated backup scheduling, plus cloud storage -- though be wary of cloud-storage that syncs but does not keep deleted files (DROPBOX), because you might delete a file, have the deletion synced, realize you NEED the file, and then be unable to get it back from the cloud.
If your data need heightened security and your campus doesn't have any suitable storage offerings (unfortunately common), look for a "zero-knowledge" cloud-storage provider such as SpiderOak or Tresorit. DO NOT half-ass your security measures! Leaking personally identifiable information has ended research careers (look up "Yankaskas" for the horror story I use in my classes).

Dropbox should keep a copy you can restore for some time.

Because no-one's mentioned it yet, I use and love CrashPlan (http://crashplan.com). It's cheap, everything is backed up and encrypted to "the cloud" and it's always running. A nice complement to full disk cloning and git/mercurial.

I have never heard of this, is it user friendly?

Very - just runs in the background silently doing its job. It's an easy and cheap and always working offsite backup.  (Note: this might sound spammy, but I'm just a happy customer)


text:
How difficult Is a PhD Program?
I am a Master's student in math who is doing pretty well so far.
I aced my first two classes (real analysis)
and my summer session in probability theory is going well, too.
Even before my Master's study, I already thought of a PhD.
Nonetheless, there are concerns:

My good grades were the result of grit, not brain,
meaning that I had to spend time to understand what the book is telling me.
Furthermore, a lot of times during exam and homework,
I had to try and fail a few times before arriving at the solution.
I did not study math during undergrad.  Before my Master's study,
I gulped down three semesters of calculus, plus LA and DE in one year.
So compared to other students, I am already behind.
In the next two months, I will learn complex numbers on my own
and review my linear algebra.
I am a professional in a field that has nothing to do with math or research.
After the holiday, I will speak to my advisor about taking a thesis class.
It's not an insurmountable obstacle,
but, in an unrelated field, I do have less time to concentrate on math.

At the end, how big is the jump from Master's to Phd?
Especially after what I said in (1), I do worry that it is beyond my ability.
I do not hold anyone responsible for my decision.
So, please kindly offer your best assessment of my situation.

I've migrated this question to academia.SE, as I think it is more appropriate here and will get better answers.

I don't think a bunch of strangers on the Internet can possibly answer this question for you. Do you know a professor in real life who will be honest with you?

My good grades were the result of grit, not brain — Do not listen to the Impostor Syndrome! Everybody has to try and fail a few times to truly learn anything.

Coursework is one thing; doing a PhD is about doing research. You can be great at coursework and not be a good researcher, and vice versa. Don't let struggles with coursework make you dispirited about doing research.

But why would you like to do PhD? Do you have a possible topic or research area you are particularly interested in? If something fascinates you and you work on it with pleasure it cannot be beyond your ability.

Difficult enough that you won't really understand until you do it yourself :)

I wouldn't worry about abilities. Maybe it is not the right question: the right question what do you want to do (after)? Do you need a PhD or helps to you? Be careful: a PhD doesn't guaranteeing you a job or higher payment.

I just wanted to add, having successfully completed elementary calculus, linear algebra, differential equations, and working on graduate level real analysis: you would be very well qualified as the math guy on many applied research teams. Maybe consider a PhD in some other STEM field that appeals to you rather than competing against pure mathematicians?

I'll answer your title question and sub-questions in three words (then add to that): Difficulty is relative.
Noam Elkies finished his PhD in math at the age of 20. The average graduate student would probably take anywhere from 4 - 5 years. That doesn't mean Noam was a god among graduate students when he was a graduate student, and that's the reason he was able to finish his thesis faster than Joe Schmoe (though he is quite exceptional) and Joe Schmoe should just give up. Far from it. The fact that you're doing well in your master's courses is a good indicator you're well qualified for the coursework part. Working on a PhD thesis means producing original research. Producing original research means dealing with not so cut-and-neat problems like ones you'd encounter in your homework or textbooks. It means working with problems that you're not sure are close-ended, open-ended, or neither. In other words, different skill sets are needed for research in addition to the problem-solving heuristics which you've already acquired from your coursework.
Since you've never wet your foot it seems to me you have no indication of your ability to do research . So what do you do? Just go for it! You only live once as far as we know, and it seems to me that your real problem is your self-imposed doubt. Get that out of your head. Research is about staring and thinking about a problem long enough then coming up with an answer which turns out to be completely wrong and in that, the process repeats until you get your "Eureka!" moment.
You should also stop worrying about others being ahead of you. There'll always be someone faster and better than you (unless your name is John Von Neumann :-) especially in academia. A good algebraic geometer from UGA by the name of Roy Smith didn't get his start in advanced mathematics until his late 20's, after working in industry as a meat lugger. He received his PhD at 35. Over 30 years later, he's still going strong. What you should learn from him is if passion calls for it, then there should be no stopping you. If you fail at a career, then that's that. It's not the end of the world. But if you never bother to even try, then you'll never know and you'll always have a "What if?" moment.
So just to sum all of this up: Apply to a PhD program and see what happens.
https://mathoverflow.net/questions/7120/too-old-for-advanced-mathematics/45644#45644 (if you're interested in Roy's post)

There are a few different things going on in this question, I'll try and address them separately.
First of all, heed JeffE's words about the impostor syndrome. The idea that some people are effortly geniuses (which is inevitably followed by "but not me") is untrue and distinctly unhelpful.
Moreover trying and failing a few (or many!) times before coming to a solution is the very definition of doing research!  The important question you should ask yourself is if you enjoy the challenge of struggling against a problem; if so, then research would be a great fit.
As for your background, you seem to have a plan to address the gaps in your background, and if you're doing well in your master's programme, I wouldn't worry too much about being behind the other students. Another challenge may be if you have a full-time job; doing a Ph.D. with a fulltime job is by no means impossible, but requires a some great time management skills. Anyway there's plenty of resources on this forum to help with this issue.
Finally, as for the question in the title, I would say it depends the most on what you plan to do with your Ph.D. after you graduate. I'm a recent graduate (in math) from a well-known North American school, and the quality of the Ph.D's varied greatly. I think it really depends on what you want to do with the degree. If getting a Ph.D. is the farthest one wants to go in academia, the standards for graduating are pretty low, and I think can be achieved not too strenuously. If one wants to be competitive in the academic research job market, the bar is quite a bit higher.

As Austin Mohr, pointed out, it is unlikely a stranger on the net fully answer this for you, but what you are experiencing is similar to what I experienced, so I can tell you how it was from my perspective.
Given what JeffE commented, studying through true 'grit', does involve 'brains' as you are developing and applying problem solving skills and stategies - which has clearly been effective, as you mention, you have good grades.  So, it seems you have developed effective study techniques.
Point 2 is pretty much not an issue, based on the skills and strategies you developed in point 1.
If you do an unrelated thesis, as you mentioned in point 3 - look at this way, you'll be developing the research and synthesis skills needed for a PhD.
What I found about the transition between the Masters and PhD, was that it wasn't so much of a leap, but a case of using the skills developed a lot more thoroughly.
I hope this helps.

In your shoes, if I were applying for a PhD program, I might be thinking in terms of an "ABD" (All but dissertation.)
From the sound of it, "grit not brain" will get you through your PhD courses. You likely will pass the "comprehensive" examination. And then the fear is that you will "freeze up" when it comes time to write the dissertation because you are behindhand in "natural" (as opposed to synthetic), talent compared to others.
You will have to balance these real concerns against the benefits of "taking courses" and getting as far as you can, before possibly running up against a brick wall with your thesis.
This was basically my story (some decades ago). I was "counseled out" of a PhD program by a dean who observed that I had the preparation and the brains for the program, but lacked the "spark," "thirst," or drive that would see others through, but maybe not me.
What's worse, as a history undergraduate with mostly As, I had trouble finding a senior thesis topic, and spent the first part of my senior year wondering if I would be able to graduate, and thinking about writing a "trivial" paper that would earn a C. In the middle of the first semester, I stumbled on a viable topic that earned me departmental honors.
I can see one of three outcomes for you. 1) A "light bulb" will go off in your head at some point before you approach the end of the PhD program. If this happens, you're fine. 2) You will somehow squeak through, not quite knowing how you did it, and come out of the process a bit shell-shocked. 3) Neither of the above will happen, as you feared, in which case you might "bail out" as an "ABD."
I can't predict the outcome for you, but am sharing my experience. Just understand what the risks are, take into account your circumstances, and (hopefully) make the best decision for you.


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Regarding "discussion sections" for undergraduate science courses
I will be undertaking my first teaching position as a graduate student in the fall - I will be leading two of the discussion sections for an undergraduate biology course.
What sorts of things should I expect to teach in a discussion section for a science course and how should I teach them?
When I was an undergraduate and took the same course (same material, different university), the graduate students had to re-teach the lectures in our equivalent of discussion sections, as if it was brand new material because the professor was not a very good lecturer.  I know the professors for whom I will be teaching these sections are rather good lecturers, so I do not expect to have to re-teach everything to the same extent.  I have several concerns, though:

These sections are about twice the size of those at my undergraduate university, and I would like to encourage regular class participation of some sort, to make sure students stay engaged.
The material is undoubtedly challenging so I am wary of bringing in too much new material, such as discussion of scientific papers, which is the first kind of discussion that springs to mind.  (Also, someone will inevitably ask "is that going to be on the exam?")

Because the material is challenging, I could probably do with some review. But how to I keep a large number of students engaged (and participating) through repetition of lecture material? Ostensibly, the reason for having discussion sections is so that there is actual discussion.

Have you asked the instructor?  (The correct answer to "Is this going to be on the exam?" is always "It is now!")

Echoing @JeffE's comment: what is expected by your manager of these discussion sessions?
The term "discussion" to me suggests something like a tutorial, where there is a significant amount of student input in the session and is not just a rehash of the lectures.
To my mind, a discussion session is an opportunity for you to show - amongst other things - how the subject material relates to the "real world", e.g. showing an example of how the particular technique/analysis/method is used. In other words, answering the prennial student question, "Why are we learning this?"

I haven't yet spoken to the instructors, but I did speak to a former student and looked at last year's material: the discussion sections look like review plus problem sets, which I think relates to the point made by @Nicholas.  I also found out that we are having a seminar during our university-wide teaching orientation specifically on what to do in discussion sections for science classes, so I think I'll get answers there. Thanks to both of you!

@dd3 once you attended that seminar, please remember coming back here and answering your question.

I can say with 100% confidence that it will depend on the professor(s).  That being said, you should expect to:

Reteach all content.  You won't actually have to, but you should of course be able to.  Students often use discussion sections to ask questions they don't want to ask the prof for fear of "looking dumb."
Give meaningful practice problems and explanations the students should familiarize themselves with for an exam.
For some classes, actually lead a discussion of topics.
Just use it for office hours.

When I taught discussion sections (for calculus, at a large U.S. state university), the students always appreciated it when I worked homework problems at the board for them. (The homework sets were large enough so that we didn't get through nearly all the homework.)
My experience is that with discussion sections there is a little bit of a good-cop/bad-cop game going on; the professor is the bad cop and you are the good cop -- embrace and enjoy the role. Help them with their homework. If appropriate, look up the professor's old exams, find out what they are like, and work out similar problems in discussion.
In short, help your students get through the course.


text:
What is a correct way for "marketing" non core PhD achievements on a CV?
Reading this question and its answers got me thinking, once non-core PhD skills are learned and perhaps mastered, how can we make these achievements clear on a CV when applying for academic positions?
For example, my PhD majors in atmospheric physics and photobiology, but a significant part of it is in Android programming (the basis of the 3 published papers so far) and a few other skills not directly related to the major disciplines, but important nonetheless.
Related, would having peer reviewed published papers be considered as proficiency in the non core subjects without taking away the focus from the major disciplines?

I see no reason at all to leave any reasonable skill or ability off of your CV. Play a musical instrument? Won some medals in a sporting event? Program an android app? Have a black belt in a martial art? Add a section called "Other skills" or "Personal Information" and briefly list these skills. There are two reasons for this. First, it can help a potential employer think of you as a real person, perhaps someone they would like to know. Second, many skills can take years of dedicated practice to master; if you can master one such skill, then there is a good chance you can master whatever new skills may be required in the new job.
Here is an example from my life. Many years ago I programmed a video game (one of the old "text adventure" style games. I placed this on my CV, not thinking much about it. It turned out that one of the elder faculty conducting the interview had played that game and remembered it fondly. This alone certainly did not "get me the job," but anything that can smooth the way can be helpful.

What you put on your CV should be tailored explicitly towards the type of job you're applying for.  If you're applying for a faculty position at a research university, putting Android programming would be a waste of time, but putting a significant grant you won would be appropriate.  Likewise if you're applying to a position where teaching is going to be a large part of your workload, put as much about your teaching experience as you can. If you are applying for a quant position on Wall Street, you need to beef up the programming skills.
So really, the answer is, tailor your CV to the job you are applying for, and given that you are applying for academic positions, you should probably leave off the non-core skills altogether.  That's not to say they aren't important, but they aren't important enough to list on your CV.  If the situation presents itself during an interview, that's probably the place to mention it.

Related, would having peer reviewed published papers be considered as proficiency in the non core subjects without taking away the focus from the major disciplines?

I'm not 100% sure what you're really asking here, but peer-reviewed research should be listed on your CV unless it is in a completely unrelated field than the one you're applying for.  Don't list the paper you happened to get published in the English Literature Journal if you're applying for positions in photobiology*.
*Unless you happen to have written a piece on the writing of 18th Century biologists and how it affects current photobiology trends.

+1 for the great answer and especially for "Unless you happen to have written a piece on the writing of 18th Century biologists and how it affects current photobiology trends."

If you're applying for a faculty position at a research university, putting Android programming would be a waste of time — Unless you are applying for a computer science faculty position, of course.

I strongly disagree with the last sentence.  List all your peer-reviewed publications in your CV.