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I am looking for professional help to verify some results I am working on and maintaining in a write-up. However, I can't get anyone for free since I was forced to leave my old university, Case Western Reserve, (due to disagreements) and I don't know if I can find an online program that assists undergraduates in their research (I should have graduated my senior year with a bachelor's degree but am technically a freshman. I have studied rigorously up to differential equations; It's complicated). Moreover, professors are busy and would most likely ignore my requests no matter how much money I offer.

Therefore, is there a way to pay for professional research help? How can I receive feedback on what I am working on?

Details: My field is mathematics. I think the work may be complicated enough that graduate students won't be able to rigorously answer everything. I need someone with a Ph.D. on the level of professors. (Technically, I am working on developing a finitely additive measure and sum that takes the average of functions defined on countably additive measure zero sets. I am looking to verify if my measure and average on a function gives in fact what I am looking for. For reference, the link to my pdf is here).

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    Why not just ask in Mathematics Stack Exchange one specific question, one from near the beginning of your exposition, and see where that takes you? Incidentally, on the first page you wrote "why null sets have zero measure", which did slightly interest me (because one of my main interests is in various ways that sets can be small, and large in the sense of being co-small), but even after a couple of minutes quickly scanning the next few pages, I couldn't find what you mean by "null set", a term whose meaning varies a lot in the literature. (Consider this as advice for presenting your results.) – Dave L Renfro Jun 23 '20 at 16:23
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    have zero measure for countably additive measures --- For ALL countably additive measures? (google "universal measure zero") If so, isn't "each null set has zero measure" automatic from a basic logic principle? I'm assuming "zero measure" refers to Lebesgue measure, which is a specific example of a countably additive measure. Anyway, this isn't the place for such a discussion. I recommend trying to get reinstated in Mathematics Stack Exchange and avoid doing whatever it is you did that annoyed the moderators. – Dave L Renfro Jun 23 '20 at 17:10
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    I'm not saying you are a crank, but this might still be relevant: academia.stackexchange.com/questions/18491/… – henning -- reinstate Monica Jun 23 '20 at 19:02
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    I really, really encourage you to seek professional mental health counseling. People aren't kicked out over quote-unquote threats willy-nilly. – Azor Ahai -him- Jun 23 '20 at 23:42
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    @Arbuja I can see you are really frustrated. It would appear your overall issue with interacting with others on SE is lack of clarity in your writing. I would recommend you slow down - break down every chunk that of content that you don't understand - and ask individual, very precise questions about these things once your account is un-suspended. Doing this will improve your ability to write clearly and think critically about the work you're interested in. You will likely answer some of your own questions that way, as well. – HFBrowning Jun 24 '20 at 21:13
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The harsh answer: You probably can't. At least, unless you are somehow quite wealthy, and if that was the case, I suppose you could simply pay for enrollment at another university. Let me try and clarify a bit.

As a researcher I get many requests from people who want me to look at their particular problem. This ranges from

  • Fellow researchers who want to discuss - I most often oblige (at least if I have time, and their request is interesting or relevant).
  • Students who I don't know in advance, who wants guidance - I help if their question is well researched and not too polemically phrased.
  • Students from my own institution, not under my supervision - I always help, as this is my job.
  • Amateurs who have invented the a new method or theory for something, and wants it checked - I rarely do more than skim it.
  • etc. etc. etc.

You are in the last category. If you are to convince me that I should prioritize your request over the others, then I should be immediately convinced that your idea is really something special. My field is not mathematics, so I can't tell here. But judging from comments from professional mathematicians, neither can they. The second option you mention is money, and this is where I think you need to be fairly rich. If a purely monetary argument should convince me to push you to the top of the stack, you would need to pay me the equivalent of my going rate when doing freelance work, around 200 €/hour.

My best bet is to either:

  • Try and get re-enrolled into a university. There people get paid to answer your questions.
  • Try and find someone who would be really interested in your specific problem, and write them.
  • Undo whatever you did to get banned at math stackexchange, and try again there. Being nice gets you a long way, being polemic and insisting that only a professor can answer your question, will get you few friends.
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If you are describing research that you are doing, then you may just go ahead and contact a professor who does the research in the field you want. Outline the scope of the work that they are required to do, and how much you are willing to pay. You'll probably need an MoU that states how any results derived from this work are to be published, and who retains ownership.

I am not in mathematics but in my research days we received requests from private individuals not uncommonly. Usually those individuals were representing themselves as entrepreneurs and they worked with us through their company, but if my advisor were to get a request to do research with potential rights to publish, and it would bring in some funding commensurate with the amount of work, we would take it whether it was a company or an individual.

So, to your point of "professors are busy and won't listen no matter how much money I offer", I think that's not true. However, don't waste their time. Presuming you are in the US or Canada, a professor can make up to and exceeding $100 per hour. You need to realistically assess how much work they need to do and then make it worth their time. If they need to do a few hours worth of work and will be paid $500, either personally or to their research accounts, that might not be very attractive. If you are offering a few thousand for a collaboration that doesn't take up their entire working life, you are probably going to get more responses. If you just want someone to check your work for like $10 and a high five, you don't want the person who accepts that offer to help you.

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I'm surprised this hasn't been mentioned yet:

If your work is at a research level (i.e., not just homework), submit your work to a journal or conference to be peer-reviewed.

Generally, the way researchers get feedback on their work is through the peer review process. The process is: (i) You write your work up properly, explaining your contribution in the context of prior work. (ii) You submit it to a journal/conference for which it is in scope. (iii) It gets assigned to qualified reviewers. (iv) The reviewers read the paper and write up their comments. (v) These comments are sent back to you along with a decision. For some reason, researchers will perform this service for free, but it may take a while.

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    For someone without any experience to submit a paper to a math journal is a fairly sure way to be desk rejected (i.e. the editor takes a glance at the paper, notices it is not written according to professional norms, to the point where it's almost impossible to figure out what the author means, and sends a refusal email with minimal to no feedback). The problem is that a person without professional training is very likely incapable of completing your step (i): writing papers is a skill, and not an easy one. – Denis Nardin Jun 24 '20 at 10:31
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    If the author needs a basic reality check on whether their research is valid at all, it is not ready to be submitted to a journal and is just wasting time of the editors. – Morgan Rodgers Jun 24 '20 at 18:39
  • @DenisNardin Journal/Conference peer-review is the standard process for getting feedback on research and a thorough answer to this question should mention this avenue. I agree that desk rejection is a possibility, but in that case it's essentially the same feedback you'd get from other avenues. – Thomas Jun 25 '20 at 0:20
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There are degree-level maths tutors around on the Internet. I'd approach one of them working in a related field and see if they're willing to work in this way. They charge say $35-$120 per hour and it would be reasonable to pay them for the time they spend outside of lessons reading and assessing what you've done.

I think a lot of tutors might be reluctant because the costs could quickly mount up without the student appreciating how time-consuming it is, leaving the student with a bitter taste in their mouth - and no tutor wants that.

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  • Someone would have to put a really low value on their expertise to do something like this for $35/hour (or, more likely, does not have the required expertise). Anyone who would do it for anything close to this price, you would be better off not bothering getting their opinion. – Morgan Rodgers Jun 24 '20 at 21:35
  • @MorganRodgers I agree the price I originally gave is at the low end. It depends if you charge every hour you work. $35/hr gives $1400 a week, which is only $70k per annum. But within about 30 seconds of reading your comment I could find a degree level maths tutor for this price: tutorful.co.uk/tutors/7vpnw2k2 – samerivertwice Jun 24 '20 at 22:02
  • @MorganRodgers bear in mind a lot of the tutors on that linked site are expecting to be paid the rate they quote for the actual lesson time and will have to do additional unpaid work outside of that. The price I indicated is assuming you also pay them for the out of lesson-time work they do too. – samerivertwice Jun 24 '20 at 22:06
  • 1: There is no way to possibly earn for 40 hours/week tutoring with out putting in around 70 hours of work. 2: This person is probably good at tutoring you for undergraduate-degree level courses. However if they have not published in the right area, they will likely not be qualified to give good feedback on research (and honestly even then, if they have only published with an advisor, they still might not be). – Morgan Rodgers Jun 25 '20 at 2:52
  • 3: this person charges really close to $45/hour since the pound is really week these days, but they charge this for tutoring a calc/linear algebra class. It is not the same thing at all as trying to give good feedback on research. Doing this is a real pain in the ass, especially if you are dealing with someone who might be a crank. I would be VERY surprised if someone capable of giving even remotely helpful advice would be willing to do it for less than $100/hour. They don't seem to be looking for a tutor. – Morgan Rodgers Jun 25 '20 at 2:54

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