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Articles by Alfred Yerger
When has one sufficiently mastered an area of mathematics?
I'd like to throw some words around, as an undergraduate myself. I don't really have an answer per se, but I do have some thoughts on the question based on my experiences and understanding so far. I think it's kind of a question you ask yourself, and answer yourself. There is not one exact standard. I expect that in mathematics, no matter how far you go, you will always be learning new things, even in the most elementary sense.
What really is mathematical rigor? How can I be more rigorous?
I'm an undergraduate mathematics student who has received some constructive feedback from two instructors at the end of my exams. Namely, that I am a bit hand-wavey and not always very rigorous. While I greatly appreciate this feedback since I intend to apply to graduate school, I worry because when I hand in assignment, I generally feel the proofs are rigorous already, and I'm not sure what I'm missing.
Legality of Universities Posting Sections of Books Online
First of all, copyright infringement (in common law countries) is not illegal but unlawful. The difference between these two things is that illegal behavior can lead to punishment, whereas unlawful behavior is simply not protected by law - if you are doing something lawful, you are protected from someone seeking redress. If someone seeks redress from you because you have distributed their copyright work, you cannot argue that your behavior was permitted by law.
Is there any way to simplify the following expression into a non-radical form?
The function $$f(x_1,x_2,y_1,y_2,z_1,z_2)=\sqrt{x_{1}^2-2x_{1}x_{2}+x_{2}^2+y_{1}^2-2y_{1}y_{2}+y_{2}^2+z_{1}^2-2z_{1}z_{2}+z_{2}^2}$$ is not a polynomial or even a rational function (over the real numbers). Suppose it was a rational function. Then $$g(x)=f(x,0,0,0,0,0)=\sqrt{x^2}=|x|$$ would also have to be a rational function - of a single variable. So let's write it as $g(x)=\frac{P(x)}{Q(x)}$. Observe that $Q$ has no (real) zeros, so it is either always positive or always negative.
What is wrong with this proof
The fundamental problem with the proof is that it mixes two proof techniques in an invalid way. The first of these is generalization and goes as follows: The second is case analysis and goes like this: The fake proof tries to mix these two motifs, such that the first half of the generalization is outside the case analysis, whereas the second half is inside the cases (and appears twice).
What is wrong with this putative proof?
The fundamental problem with the proof is that it mixes up two valid proof techniques in an invalid way. The first of these is generalization and goes as follows: The second is case analysis and goes like this: The fake proof tries to mix these two motifs, such that the first half of the generalization is outside the case analysis, whereas the second half is inside the cases (and appears twice).
Advice on finding counter examples.
It seems like people have already come up with answers regarding your particular problem, so I think I'll just throw a few words around regarding how to find counterexamples. I have two basic strategies myself that I use when thinking about theorems and problems. 1) Consult a list of known counter-examples. This is a really good strategy in subjects like analysis and topology. I don't know how well it works in algebra, but I expect there are several known pathological counter-examples.
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