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    Question on vector space

    So I was thinking and I was wondering if we could have a set of vectors that spanned just the unit sphere, and nothing else beyond that. So, if we replace euclid's 5th postulate to give us spherical geometry, a line is a circle on the surface of some sphere. If we have two perpendicular vectors...
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    Will AI ever achieve self awareness?

    Wow! There have been some good posts here. Let me give a quick thought expirament. It is known to be possible to to computer stimulations of various phenomenon. For example water going into a container. What is done is programming Newtonian physics into the computer and seeing what happens with...
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    Analysis class

    Since this is about my analysis class, I think this is the right place to put this. So, I'm a couple weeks into my class in Mathematical Analysis, and the whole class is a cure for insomnia, and I don't even have insomnia. The textbook my professor assigned looks like they liked baby Rudin, but...
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    Analysis Principles of Mathematical Analysis by Walter Rudin

    It's a good book, and it's meant to be read in a linear fashion. Don't complain about it if you skip parts of it. Oh, and for the prerequisites you don't need any of that nonsense. If you understand the rational number system, the binomial theorem, and some previous knowledge of trig functions...
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    How to address professors properly as you continue as a PhD student?

    If they don't want to be just called by their first name, they should honestly sign what they want you to call them. If they don't like that you called them what they signed as just put "Dear sir or madam" in the next email as your opener. Or you could say that a name is just a reference, what's...
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    Recommend a Rigorous Calculus Book

    Calculus and rigor do not belong together. Baby Rudin should suffice for all your calculus needs. It has problems that require a deal of ingenuity that even Rudin admits in the preface. I think that qualifies for your examining thoroughly from every angle. I will tell you this, if you understand...
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    News Iraqi unrest, Syrian unrest, and ISIS/ISIL/Daesh

    You're not saying ISIS wants to go after anything outside of Iraq, are you?
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    What web browser to use?

    If you want a secure browser, Lynx is perhaps the best choice, but that might be too secure for you. I always say that the only purpose of internet explorer is to download other web browsers(ie usually comes with windows). I wouldn't trust ie for security and I wouldn't trust google to keep its...
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    High Level Language

    Thank you guys. @DH Wouldn't you need to understand all these low level machine constructs if you were to make a high level language. It just seems that abstracting away for lower level concepts will cause your computer to not run as efficiently as if you ran everything on lower level concepts...
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    High Level Language

    I'm looking for resources that discuss how a high level language works from a fundamental level. I'm not looking for something that just says if you put this in your program, this is what will happen. I looked at your programming resources page, and that is all it gives lend to. I'm looking for...
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    Finding Cardinality of Power Set

    The identity map is from P(A) to S. Anyways, if we define f(a)= 1 if a\inC and f(a)=0 if a\notin. Then, \forallC\inP(a) and \foralld\inS, we define CRd if f(a)=d(a). Wouldn't it suffice to show that the relation R is a bijective function?
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    Finding Cardinality of Power Set

    @LCKurtz, I didn't assert that it is just a bijection, but that it is an identity map. Perhaps, this is not the standard definition of a subset, though.
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    Finding Cardinality of Power Set

    Well, could we just define a subset of A by assigning a "0" or "1" to each element aεA? Then the set of all subsets would be the set of all functions from A to {0,1}.
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    Finding Cardinality of Power Set

    Homework Statement Let S be the set of functions from a set A to {0,1} Prove that |P(A)|= |S| Homework Equations P(A) is the power set of A The Attempt at a Solution I have no idea how to do this... If A is finite then A has n elements, and we can write out the elements from one to...
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    Importance of Binomial Theorem

    The binomial theorem can be of great help in analysis if you are trying get some estimates, and you have something in the form of (a+b)n. I know it's used several times for this reason in baby rudin. Practically, it can be a lot faster to evaluate the coefficients of each term then by just...
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    Finding range of f(x)

    Well, the range is just R. No? Unless you are talking about the image of f. By the way, I do not believe that f is uniquely defined, as haruspex pointed out.
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    The Enduring Debate

    LOL!! Unless you're the grim reaper!
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    News Finland's hockey fans must be going wild

    I bet Putin's not too happy about this...
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    Understanding Biology

    Well, you don't need the derivations of the physics formulas. They would be fine if the physics worked exactly like the formula says. There's no need to understand what they mean, they are mathematical statements that will give true predictions. However, we could equally claim that the migration...
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    In the complex number system, why can't 1+1 = 0 ?

    The problem is that for complex numbers, √(a+bi) has two complex roots. With real numbers, one is positive and the other is negative. We can assign the positive one to equal the positive square root. However, with complex numbers it isn't so simple. A complex root is neither negative nor...
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    Beat the speed of light

    In my personal experience, "why does it happen?", and "how does it happen" are physics questions. "Why does it happen?", is usually a metaphysics question. Perhaps you should read some Aristotle for that. Actually, an object can go faster than c. If you're talking about an object in a...
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    Understanding Biology

    So, I have a biology and physics question that I don't really understand. I go to physics class, and they talk about atoms, and particles, and what we can do with them. I understand that. Then I go to biology class, and they start talking about cells, DNA, RNA, and they talk about individual...
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    After calculus first pure math course, having a hard time.

    Proof by contradiction sounds like a great way to attack some of the problems you listed. Unless you're a constructivist. In number theory, induction is a very important tool too. For the first one, why can't a divide b if a>b?
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    Not understanding textbook solution: Mary Boas mathematical methods

    They want you to write it out using the binomial coefficients. n!!=1*3*5*...*n where n is odd.
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    Our Dependence on Technology

    Chronos mentioned A Clockwise Orange. We should probably be somewhere in between using no technology and completely depending on it. I advocate neither. We usually need to think about how the technology works to improve it, instead of treating it like magic. Debit card is another good example...
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    Our Dependence on Technology

    My personal opinion is that we shouldn't be dependent upon technology to think for us. Good technology, on the other hand, makes life easier for us. For example, a dishwasher gives us more time to do other stuff. Or a program let's me calculate gigantic matrices. The problem is that technology...
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    Mathematics Research

    Okay, so I'm new here, but I have a few questions about research that mathematicians do. Firstly, what is it like to do mathematical research, and secondly how would I go about doing that? Currently, I'm just an undergrad in mathematics and my highest level of knowledge is just complex analysis...