# Tag Info

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It means they are exactly the same except for the names of the elements and the name of the binary operation. An isomorphism between groups is a function that renames all of the elements. (Hence, it is bijective... each element in the first group gets renamed to be exactly one element in the second group.) The reason we care is that if you are only ...

13

In loose terms it means you can't tell them apart. They are the same except that the elements have different names. For example the group $Z_2 = \{0,1\}$ with the obvious rule for multiplication is isomorphic to the group {even, odd} with the usual rule for addition. @Dorabell 's confusion (see his comment below) was my fault for calling the operation on ...

9

When we say that two groups are isomorphic, we are saying that they have the same structure and invariants as groups. An isomorphism between two groups do more than matching elements: it matches subgroups, normal subgroups, characteristic subgroups, conjugacy classes, $p$-subgroups, Frattini groups, ... In other words, two isomorphic groups can be ...

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Some things to consider: To a various degree this happens to all of us. It is an essential skill, to be able to leave a problem behind, even if only for a few days. It is fine to skip a problem, esp. if you have solved almost all the others from the section. If it is really bugging you, get help from math.SE. Time is precious—get things done, and get done ...

9

Find Terry Tao's blog. He talks about his experience of learning at different levels of mathematics education. Among other insights, he writes how patterns from competition problems he later discovered to be examples of more general, deep and beautiful results. What I took away from all that was that while solving competition problems isn't directly ...

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I am not one thousandth the mathematician that Terry Tao is, but my own feeling is rather different. I had a college classmate who was far better at competition mathematics than I was, and when we went to grad school (together), he seemed good at following a prescribed path, but not so good at striking out on his own. In later professional life, he made no ...

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Srinivasa Ramanujan was such a mathematician. He failed to got admitted to college but he became one of the best mathematician of $20$th century. Evariste Galois failed to enter to Ecole Polytechnique twice.

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The analytic number theorist Hua Loo-Keng overcame abject poverty, handicappedness, political persecution; for more information refer to one of his faithful biography. Charles Hermite overcame much too, but in different aspects; he failed nearly every math exam that he was to take. To supplement, the analytic number theorist Chen Jing-Run, the man closest ...

5

Being successful at Olympiad mathematics is certainly correlated with being successful in later studies and research, but there is no implication in either direction. This is what you would expect a priori: coming up with creative ideas is a part of the work of a research mathematician, but by no means the only (and arguably not the most important) part. ...

5

It means that, even though the groups contain different elements and combine according to different rules, they are nevertheless from the perspective of group theory essentially identical in every important respect. The easiest example of this comes from elementary school arithmetic: We learn at an early age that adding two odds gives an even, etc. We ...

4

What you defined could be called a valid formal proof. A valid mathematical proof (or a proof accepted by the mathematical community) on the other hand might be described as an informal(!) arrangement of arguments that the reader finds convincing in the sense that he/she strongly believes that it is possible to write down a valid formal proof reflecting the ...

4

Andrew Wiles with his Fermat proof is an example of massive struggle. He worked on the topic for 9 years and got demolished when presenting an erroneous proof after 7 years. There were several additional problems, but I forgot the details. You can read about it on Wiki and in much detail in the very accessible "Fermat's Last Theorem" by Simon Singh.

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I'm not sure how much value I'm adding here, but perhaps this example will be interesting to kids of the modern age. Suppose we have the following two graphs of equal size: Graph 1: A group of youths with Facebook accounts. Some pairs of people in the group are friends on Facebook, some are not. Graph 2: A room full of pre-programmed old-school paging ...

4

Your questions in order: Yes, it is highly advantageous to be exposed to more sophisticated ("graduate") subjects as early as one can tolerate it. Not clear that one should "study" them. Yes, some people do "read ahead". I myself found it very helpful. I would claim that the widely-believed sense of "deep and slow" versus "skimming ..." is a fake ...

4

Your question is very broad, and I'm not sure this fully addresses it; but this is too long for a comment, and hopefully you find it useful nonetheless. I think there's a couple false assumptions here. First, that there is a "better" way to approach studying mathematics. People vary wildly in how they learn, and ultimately I think it's best to find an ...

3

I would argue, as you say, that knowledge is worth something even in absence of practical applications. However, that is not to say that theoretical mathematics has no practical applications. In my opinion the following is the main difference between applications of "theoretical" and "applied" mathematics: Pure mathematics is an investment for the far ...

3

It happens with me but only when am practicing at home . During exams , the adrenaline in me kicks off and I do whatever I know I can . Then with the remaining time I approach the problems I left and fortunately I have solved all in most of my exams . But during practice , there isnt much focus . I feel sleepy if I dont get something and cant proceed further ...

3

You have solved a mystery for me. I teach mathematics in a university, to bright, motivated students. They almost all have great difficulty in understanding the concept of equality, and in substituting expressions. For instance, if you write $f(z) = \frac{e^{iz} - e^{-iz}}{2i}$ and then ask them what is $f(3+4i)$, they often don't understand that you mean ...

3

As other answers already point out, isomorphism is just relabeling elements and renaming operation, but all relations you could think of such as subgroups, quotient groups, generators etc. are preserved, up to relabeling. Let me try to draw an analogy with similar but undoubtedly more familiar concept: what does it mean that two triangles (in a plane) are ...

3

The following is just my opinion. I commend you on trying to further your brothers education but I would be careful as to how far you go and in what order. In my experience, we tend to forget the things we knew before we learnt harder stuff. Given that your brother is asking about something notational, it may be that he doesn't really understand why you are ...

2

There is a connection but to make it clear you have to be more precise about the boundaries of the inner integral on the right hand side of your second equation. The primitive function $F$ of $f$ is only determined up to a constant. Assuming that it is chosen so that $F(a)=0,$ then we have $$\int_a^bFg'=\int_{x=a}^b\int_{t=a}^xf(t)dt\ g'(x)dx$$ The ...

2

Here is Doron Zeilberger's opinion bearing on this topic with a pointer to some feedback from me (which includes some remarks on Hagen's point about Principia Mathematica).

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Not exactly an solution to your question, but frankly it's too big to be fitting a comment. The distinction as to exactly what is being linear is very important. It is possible to approximate non-linear behaviours if we are allowed to make the linear space (as in linear algebra) large enough. Approximating solutions to smaller dimensional non-linear ...

2

Comment You are right: in First-order theory of arithmetic (see : Peano axioms) a term can be : a variable, a constant ($0$) or a "complex" term built up with functional symbols : $S, +, \times$. Some examples of arithmetical terms : $0, S(0), S(x), S(0)+0, \ldots$ A term acts as a "name" for a number. We define : $1=S(0), 2=S(1)=S(S(0)), ... 2 John Nash struggled with significant mental health issues when he should have been in the twilight of his mathematical journey. He and his wife spent many difficult years battling with this illness. Slowly, Nash started to get back in touch with the mathematical community in Princeton; engaging with the students, his passion for mathematics never died. It ... 2 TLDR : if the result is more important to you, go applied. If the road-to-result is more important, go pure. I'm sure you are aware that behind many world-changing inventions lie an accumulation of "pure math" results, that seemed useless at their time. Traditional examples include number theory results that are used in cryptography, or Turing's idea of a ... 2 If you've thought about the logarithm in the context of complex variables, you know it must be defined with branch cuts or else be a multivalued function. If you want to "graph" the multivalued function, what you have is a Riemann surface. Indeed, Riemann surface theory is a natural outgrowth of complex analysis when you want to see the topology behind ... 1 This question is vague, but my general advice would be to understand what the mathematical objects and symbols you're working with mean. Then you will be able to determine for yourself whether a substitution is valid. If you're just manipulating symbols with no idea of what anything you're writing down actually means, that's when you run into trouble. For ... 1 You wrote "when is it allowed", but I think you are trying to ask something like "when is it a good idea"? If you are asking the latter, then I usually tell students something like "I want to do as little work as possible". Meaning I relate the problem to something I already know, like you did in the example. Another simple example is, I know that ... 1 Let$p,q \in M$and let$pq$be a geodesic connecting them.$pq$will be minimizing if and only if its intersection with the cut locus of$p$is empty. The cut locus of$p$is made of: (1) the conjugate points of$p$and (2) those points$r$such that there are multiple minimizing geodesics from$p$to$r$. Since, by assumption,$pq\$ contains no point ...

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