Logan M
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Member for 13 years, 10 months
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Last seen this week
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USA
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Notable math from those without math PhDs
Insofar as theoretical computer science is a part of mathematics, Chomsky should qualify. Despite being a linguist, his work helped shape modern theoretical computer science.
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Is it possible to construct a finite mathematical universe?
It seems to me that the "conventional mathematics of infinity" was started by Cantor much less than 400 years ago. Of course, there is a concept of infinity in calculus, but that's something totally different in my book.
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What are good non-English languages for mathematicians to know?
If you're going to choose a language like C which is really only useful for computational programs (and software design, but I assume most mathematicians don't do too much of that), you might as well say Fortran rather than C. It's still the language of choice in most hard sciences.
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Finding the degree of minimal polynomials
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Finding the degree of minimal polynomials
@Georges Elencwajg that's exactly what I meant. Sorry for the terrible miscommunication. I've added this to the answer, with all the hypotheses clearly stated.
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Finding the degree of minimal polynomials
In this case, the proof I had in mind is by induction on n on the stronger statement that if $a$ and $b$ are of degree $k$, $m$, for $m,n$ relatively prime, the degree of $a+b$ is $km$. I was under the impression this was a well-known result, but it may be incorrect. If it is true, then Eisenstein gives that $x^{a_i} - p_i$ is the minimal polynomial for $sqrt[a_i]{p_i}$, so $sqrt[a_i]{p_i}$ is of degree $a_i$. I suppose we need the much stronger condition that $gcd(a_i,a_j)=1$ for $i \not{=} j$ rather than just $gcd(a_1,\ldots,a_n)=1$ for this. I blame this obvious mistake on lack of sleep.
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Finding the degree of minimal polynomials
Perhaps it will be helpful to enunciate my claim fully. Suppose $p_i$ are integers, $a_i$ natural numbers, for $i=1,\ldots,n$, such that for each $p_i$ there is a prime $q_i$ such that $q_i|p_i$ and $q_i ^2 \not{|} p_i$, and additionally that $gcd(ai,\ldots ,an)=1$. Then I claim that $\sqrt[a_1]{p_1} + \cdots + \sqrt[a_n]{p_n}$ has degree $a_1 \cdots a_n$ over $\mathbb{Q}$. All the other claims I made, the "proofs" I thought I had were flawed except in the case n=2. Please disregard them.
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Finding the degree of minimal polynomials
Scratch what I said above; it's not true unless each $p_i^{1/a_i}$ has minimal polynomial of degree $a_i$. This holds in the case that some prime q divides $p_i$, but $q^2$ doesn't divide $p_i$, by the Eisenstein Criterion. With more advanced arguments, a little bit stonger statements can be made. It fails in the case where we take $\sqrt{4}$. Another similar case is that the kth root of unity $e^{i2 \pi /k}$ satisfies the polynomial $a^{k−1}+\cdots+x+1$, which is of degree k−1. But when $p_i^{1/a_i}$ is of degree $a_i$ over $\mathbb{Q}$, the above should hold.
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Finding the degree of minimal polynomials
There are a good number of cases where your statement will hold, but it's not true in general. If the $a_i$ and the $p_i$ aren't related in any obvious way, it's probably true for small values of n, but I'd still check. Wolfram alpha (www.wolframalpha.com) can compute the minimal polynomials in sufficiently small examples, and most computational algebra engines can do it for arbitrary numbers of the form you want.
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Finding the degree of minimal polynomials
I'm not totally sure what having $gcd(a_n,p_n)=1$ gives you (or conditions regarding pairs $(a_n,p_n)$), but if you have $gcd(a_1,...,a_n)=1$ then your statement holds. If $gcd(p_1, \ldots ,p_n)=1$ and $a_1= \cdots = a_n$, it should also hold. I'm not sure about the general case for $gcd(p_1, \ldots ,p_n)=1$, though I wouldn't be surprised if your statement held then.
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Approximate search space on a 5x5x5 cube with 3 different possible classes?
If all you need is an estimate, I'd run a Monte Carlo code. Something like: 1) Randomly generate a position 2) Check if it's a valid position or not 3) Check under what rotations, reflections the position is invariant. 4) Repeat Once you have a lot of trials (a computer can easily do a few billion), sum the reciprocals of the numbers from step 3 of those positions which were valid. Then estimate from this the probability that a randomly chosen configuration fits your criteria, and multiply by the number of positions (3^125), for a decent estimate.
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Use of traces in physics
I'll attempt to justify some of why traces are important, though not solely from a physics perspective. The trace of a matrix $M$ comes naturally (as does the determinant) from the characteristic polynomial $p(\lambda) = \det(\lambda I-M)$. Namely, for any algebraically closed field, it is the sum of the roots of $p(\lambda)$ counted with multiplicity (the determinant is the product). This is clearly coordinate-independent and a rather fundamental quantity. The usual definition lacks any intuition, but is more useful for generalizing to arbitrary matrix rings and for efficient computation.
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Place of Analytic geometry in modern undergraduate curriculum
From my limited experience, these types of geometry courses are not generally taught for math majors. Typically, they are more directed towards math education majors or engineers who are trying to get a math minor. Non-euclidean geometries are certainly mentioned occasionally, but are not always explored with any depth. The reason for this is that geometry isn't totally necessary anymore. It would be difficult to do anything serious without, say, knowing what a group is, but geometry has become a niche topic. Graduate students typically learn some advanced geometry, but undergrads rarely do.