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There's logic and logic! For a first math logic course on first-order logic (and I'm including here things like the completeness proof for "ordinary", i.e. countable, languages) you need no serious set theory. Sure, it's standard to use set theoretic notation and say e.g. that a model has a domain which is a set equipped with functions and relations where ...


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By now, you, Thomas Nesbitt, see this answer, you should be somewhere over 22, based on what you said and the time that has passed by. Several of the answers mention the Fields medal as an example of an award which has an age limit. So does the Nevanlinna prize. Assuming you had started to study mathematics now, add another 12 years, and because the Fields ...


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This may be useful for a somewhat different perspective. My personal feeling is that it is (I would say) of foremost importance that you become comfortable and nurture a proficiency with proofs. The topics you mention, when studied in the depth I would think you want to explore all have a "proof component." While this point could be argued, typically in ...


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I think you will probably like any of the introductory books by Rey Pastor. The issue there is that he was Spanish, so you won't probably be able to find a book by him in English. I have the three volumes of his Calculus course, and it's the most comprehensive book I've ever seen on the subject. A book I like that has a small introduction including some ...


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I took differential calculus twice at two different colleges and it still took me at least another decade before I understood it. It is a simple subject but not the way it is taught. Differential calculus is the study of one particular property of functions. So it is absolutely necessary that you clearly understand what functions are including graphical ...


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I would recommend Thinkwell Homeschool. You don't have to be a homeschooler to use it (though I am). It can be taken as a full course or used to supplement another. They offer three calculus courses: AP Calculus AB, AP Calculus BC (link in the sidebar), and plain-old "college" Calculus (I and II combined). I'm currently taking AP Calculus AB, and I'm loving ...


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Download the first edition of Gilbert Strang's Calculus ,available here and get yourself a copy of Adrian Banner's The Calculus Lifesaver: All the Tools You Need to Excel at Calculus, which you can get a used copy of at Amazon for just over 4 bucks. If you carefully work your way through both those sources, you'll be well on your way to mastering basic ...


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Stewart Calculus. Don't do Spivak, that is just too much in my opinion. Just start from the beginning of any edition of Stewart Calculus. Section by section work a good amount of problems, to ensure you are learning. Trust me you want a good base to start from in order to get into more theoretical/rigorous calculus. This is coming from someone who took ...


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I'm self studying calculus right now. MIT OCW courses are perfect (far better than my real classes). They already have notes, so you don't need to use your pen (except for problems), you can review a lot easier, and it's free. Single Variable Calculus - MIT OCW Multivariable Calculus - MIT OCW


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There are many ways you can learn calculus. There are many textbooks out there dedicated to calculus, such as Thomas' Calculus, Stewart Calculus, Spivak Caclulus (more rigorous than the other two and is meant as an introduction to real analysis), high school calculus books, etc. There is a professor called Paul and he has a lot of good notes on algebra, ...


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To vastly over-simplify things, there are two types of "difficulty" to think about: (1) Difficulty presented by the structure of the class. (2) Difficulty that you will have assimilating concepts and methods. (1) is determined by things like volume of homework, types of questions on homework, and structure and frequency of exams. Such factors vary from ...


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Number theory will be more fun and "cooler" for you. If you were worried about becoming a professional mathematician, Analysis might be more important, but if you are going into math education, I would go for number theory, and then either prob or Euclidean Geometry (which might be fairly tough if they are doing modern developments).


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Stroud would be an excellent choice, especially since it has answers to problems in the back of the book (remember: you only really learn math by doing it!). I also see that it has a section on statistics, so that may be all you need. If not, I think a good book that is tailored to your needs is something like An Introduction to Medical Statistics. Do a ...


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You might be interested in the books Algebra and Trigonometry by Gelfand. Also, it's not dry or old, but the Precalculus textbook from artofproblemsolving.com won't spoonfeed, at least. From the book description: It includes nearly 1000 problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics ...


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Yes I hear your point. Most books released these days look to spoon-feed. But that does not apply to all new books. I mean Spivak's books, Chapman Pugh's text on Analysis are examples. Now these are the books I perused during A Levels. I only got my hands on them because the government sells them for dirt cheap prices (I mean for less than 20 cents US). ...


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Definitely review! Think of this as landing a job. One you look forward to advancing your career and/or happiness, to boot. But the employer wants you to be good at something you have skill at but haven't done in a while. Do you just play up the past and wing it from there, or do you re-familiarize yourself with that skill set as much as you can? The latter, ...


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I was undergrad math-physics and did compsci on the side. Afterwards i worked a CS job for 2 years and then applied to a math masters program with a similar goal. Having now completed it, I can say that it should be doable with little prior study (I never had analysis in undergrad) so I never even prepared for it in grad school. Though I prepared nothing ...


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Let us call $k$ the division factor (i.e., $S_{j+1}=kS_j$). Considering $n$ segments generated by $n-1$ divisions, we have $$L=S_1 (1+k+k^2+k^3+k^4...+k^{n-1})=S_1\frac{1-k^{n}}{1-k}$$ which gives $$k^n- \frac{L}{S_1}k+\frac{L}{S_1}-1=0$$ On the other hand we must have $$S_n=S_1 k^{n-1}$$ Solving the system given by the two last equations yields the ...


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The first few chapters of GH Hardy's "A Course of Pure Mathematics" may be worth a read.


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As dry, old and rigorous as it gets "Advanced Mathematics Precalculus with discrete mathematics and data analysis." It's what I had in High School, although I had a modern textbook as a suppliment. There might be newer versions out, but I assume you want the older ones.


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If you have solved an open problem and can monetize it then this question is easy. For example, consider factorization. If you really developed an algorithm for efficiently factoring large numbers, you could write a paper in latex describing your algorithm and proving its correctness. You could send that paper to me. It is likely I will not understand ...


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I'm by no means someone who does research, but I don't see this huge difference. If you find the question on a textbook you have the following theorem at your disposal : "There exists a solution that employs only theorems I've already seen and it is reasonably short and relatively easy to find " This is an important factor because it keeps you determined ...


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Perhaps the question in general is more concerned with cognitive processes than with mathematics alone. Having said that, there are differences between typing (interpret it as your "blogging") and handwriting: Marieke Longcampa, Marie-Thérèse Zerbato-Poudoub, Jean-Luc Velaya, "The influence of writing practice on letter recognition in preschool children: A ...


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In mathematics, a statement is either an axiom provable by using existing axioms contradictory to existing axioms (commonly called wrong) undecidable by using existing axioms Terms like "obviously true" and "self-evident" have caused severe problems in the history of mathematics and philosophy as a whole. Thus, they lead on a dangerous path and should ...


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Hardy to Ramanujan: 'Let me put the matter plainly to you. You have in your possession now 3 [Ramanujan had only two] long letters of mine, in which I speak quite plainly about what you have proved or claim to be able to prove. I have shown your letters to Mr. Littlewood, Dr. Barnes, Mr. Berry, and other mathematicians. Surely it is obvious that, if I were ...


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I think that there are three things you need to do. Type it in $\LaTeX$. No matter how good your results, or how clear your exposition, if it is written in Word using Equation Editor then noone will take it seriously and it will be lost to the world. Get the source-code of some good papers off of the arXiv to see how to do this well (find a paper, then go ...


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I'm no expert on this subject, but the impression that I have comes down to this: your work may be very worthwhile, but it's hard to be taken seriously if you cannot communicate your discoveries. A particular difficulty in getting published in mathematics is that the subject is very technical -- to use a shaky metaphor, mathematicians speak their own ...


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There's a list out there that I'm sure someone else will be able to attribute that assigns crackpot points to a paper to determine whether or not it's worth reading ( I think John Baez? ). Explicitly or not, most professionals decide whether or not to read a proof based on the ideas in that list. At any rate, regardless of what you've proved, and ...


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If an amateur has created a proof of a problem the natural step is to publish it on either arXiv (or if you don't have access) viXra. At that point you have documented you were the first to get the idea at a certain time. Now if you think it's material worthy of appearing in a paper you should attempt to reach out to relevant journals (which will have a ...


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For every chapter after the lecture you should sit down and try to attempt the problem sets. Usually there will be an easier problem that the instructor has gone through in class that will lead you through most of the exercises. When you get stuck go back and reread. Find examples in the text chapters, which you also should read and work through. ...


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Many schools offer at least some free academic support such as tutoring or walk-in help sessions for their classes. You might investigate that possibility at your school. (It sounds like you don't want to 'invest' in a tutor.) As for your bigger question: should I drop maths altogether...you might want to ask yourself questions like these: (1) 'Why am I ...


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To all the people that tells me: "I was always bad in maths" I answer: "That's because you were not ready to invest enough time for it" Improving your efficiency while learning is matter of experience, and sometimes tutor can help (in order to discipline yourself for example).


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Here's my take, in diagram form: The diagram is organized from basic (top) to advanced (bottom). A solid arrow indicates a more or less definitive prerequisite. For instance, I consider numbers and sets a prerequisite for both groups and discrete math because working with sets is essential in both subjects, and because numbers and sets are a good place ...


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You pretty much need #3 and #7 before anything. Which order you do these in doesn't matter. A lot of stuff on your list will require multivariate calculus, which is why I say #7. And in #3 you will build up your mathematical maturity/proof skills. Then a whole world will open up to you, so you could do whatever you find interesting. I wouldn't recommend ...


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I suggest the following sequence: 3-5-2-8-4-7-1-9-10-6-18-11-14-12-13-16-15-17. Exact contents and prerequisites of your material may call for some permutations, but this shouldn't be too bad. The list of topics seems good to me. As you proceed, you will develop a taste that will tell you what you want to focus on and what requires clarification. When ...


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How much mathematics should a student of mathematical logic know? A student at what level? At introductory and intermediate levels, you need little specific mathematical background, just the ability to follow mathematical proofs. And in general, I'd say that you can get quite a long way studying various areas of mathematical logic picking up what you ...


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You should be well versed in Ordinary Differential Equations, Boundary Value Problems, including Eigen Value Problems. Solving problems is very essential. You should also have studied a course on calculus.


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Besides calculus you should know complex numbers, trigonometric functions, Euler's formula. You can learn other concepts like sinc function, distributions, special functions (top hat fynction, dirac-delta etc.) while studying. If your calculus is good it is not that hard.


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Its been a long while now but I know I would go to your book store by the Engineering books youll find a sparknote cheat sheet section pick up calc 2 & 3 as well as the diff eq. It is the best way to cheat the foundation to keep up with the class. 3d coords and conversion of them ie polar. Vector. Scalar. These were necessary to see id sets for ...


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For a general algebraic structure ("universal algebra"), one defines a congruence as an equivalence relation compatible with all of the operations of the algebra, i.e. $\,a'\equiv a,\ b'\equiv b\,\Rightarrow\, a'\circ b'\equiv a\circ b,\, $ for all operations $\,\circ\,$ (just as for congruences on integers). Such compatibility implies the algebraic ...


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I presume when you say "equivalence $a \cdot b^{-1}$" you mean a relation $a \cdot b^{-1} = 1$ in a group presentation. (We may also consider a more general situation in which we impose the relation $a \cdot b^{-1}$.) It is not clear from your question wheter $a$ and $b$ should be generators or arbitrary expressions, but that doesn't matter too much. In ...


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I presume you mean the equivalence relation induced by a subgroup $H\le G$ defined by $a\sim b$ if and only if $ab^{-1}\in H$. I have two answers for this. (1) Groups are meant to study symmetry. I think groups are to group actions as potential energy is to kinetic energy: groups act on things. So group actions are of fundamental importance. Often we ...


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"How to Prove It" (Velleman) is a good introduction to mathematical proofs. "How to Solve It" (Polya) is a good introduction to the methods of problem solving. "Concrete Mathematics" (Knuth, et al) is a good introduction to discrete math. "Language, Proof, and Logic" (Barwise, et al) is a good introduction to formal logic (with Fitch) "A First Course in ...


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It is increasing (not necessarily strictly) because you are taking supremum over an increasingly bigger set.


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As an engineer interested in mathematics, you might want to look into the field of Continuum Thermomechanics. There are (applied) mathematics departments which offer such courses; yours might be such a school. Since you mentioned that you have done some self-study, books to look at as an introduction include: 1) The Mechanics and Thermodynamics of ...


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Aren't engineering students supposed to know algebra and calculus? While mathematics is certainly a better major than, say, the study of 15th century Armenian literature (which is fairly useless as far as today's jobs are concerned), you would be better off studying something with the greatest likelihood of landing you a good job after you graduate. In that ...


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You're probably in your early 20s, so: relax, you're okay. There's plenty of time to learn math. You have not missed some sort of mandatory train for becoming a mathematician or even just doing math at a reasonably advanced level. College is a great time to start doing mathematics. If you finished Spivak in just two months (depending on the detail you went ...


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For reference's sake, here is Chern's original paper: http://www.maths.ed.ac.uk/~aar/papers/chern7.pdf The "modern" version of this theorem can be found in Liviu I. Nicolaescu's personal website, Lectures on the Geometry of Manifolds, Chapter 8. I have not read his proof, so cannot comment on it. Milnor&Stasheff's bilbliography listed a few other ...


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Having been a securities analyst and an actuary, I might give you my perspective (based on several years ago, having long since retired). The math component in A.S. does not emphasize rigor. Rather it accentuates proficiency in speed and accuracy in applying certain concepts that are "somewhat" relevant to the actuarial profession. In order to pass exams ...



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