1,300 reputation
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Ask Me About System Design. I am willing to do email correspondence on the subject.

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__Interests: General Algebra, Computability, Enumeration, Prime Gaps

__Project: Bounds on Jacobsthal's Function

__Project: Combinatorics of P's Ring Toss

__Have: copy of Erik Westzynthius' (Only?) Paper

__Want: symbolic dynamics on infinite directed sets, esp. forests

__Contact: through Will Jagy, or guess the following Hangman dro_d__r_ard__ma_l.com

(Please do not merge keep me, or even merge me: merge keep 3402 instead)


22h
comment SHPS and SPHS inequality using monounary algebra
@Arturo, apologies for the misrepresentation. Hopefully your reasons for degree of participation are cleaner than mine. I am glad to see your writing on MathOverflow. I hope I am not wrong in my estimation of universal algebraic knowledge of the present math.SE participants. Gerhard "Will Use A Different Tense" Paseman, 2014.04.17
1d
revised Advice for number theory library
added 21 characters in body
1d
comment Advice for number theory library
Thank you. I will fix the i/y problem. You are welcome to modify the rest and remove the parenthetical comment. (I've never studied the relevant languages, and prefer poor Romanization by myself or correction by someone else more knowledgeable.) Also, copy-paste has given me problems lately. Gerhard "May Switch Back To Pencils" Paseman, 2014.04.17
1d
answered Advice for number theory library
1d
comment SHPS and SPHS inequality using monounary algebra
Arturo and many others familiar with this level of universal algebra often participate on math.stackexchange. If you @anurag decide to ask similar questions on MathOverflow, I recommend prefacing with a little background, e.g. "I'm a third year undergraduate with a little linear algebra background trying to go through paper X, and teaching myself universal algebra out of text Z. How do I approach Y?" It will then be clear that you aren't a graduate student asking others to do your work, and you will be redirected more gently. Gerhard "With Less Downvotes As Well" Paseman, 2014.04.17
1d
comment SHPS and SPHS inequality using monounary algebra
It is more a matter of exposition. Now that I have reread it, I find less problem with it. If I were concluding with A and explaining it, I might go a little more slowly, or indicate with more clarity how to produce A_4 union A_1. My initial (mild) objection stemmed from the impression that you were taking a homomorphic image of a subalgebra of A: you aren't, but it is easy for me to get the wrong impression, and I have seen this material before. I can't think of a good way to improve the exposition though. Gerhard "Not To Mention Other Students" Paseman, 2014.04.17
2d
comment SHPS and SPHS inequality using monounary algebra
You should claim a subhom of A is isomorphic to A_4 union A_1. (Although I think I see that the union is also in H(A).) Also, I suspect this question and answer more appropriate for math.se, in my humble opinion. Gerhard "Ask Me About System Design" Paseman, 2014.04.15
2d
comment Eigenvalue of (0-1) matrix
If you really don't know how to proceed, I recommend posting this on a different forum. Math.stackexchange, ArtOfProblemSolving, and other fora have people who may give you more hints than I have. Good Luck. Gerhard "Confident You Will Solve It" Paseman, 2014.04.15
Apr
15
comment Eigenvalue of (0-1) matrix
I could, but I want to make sure that you do the work. Again, consider the (nonzero) eigenvector $a$, and assume that it has eigenvalue 3. Why is this inconsistent with the posited matrix A? Again, using the component of $a$ of largest absolute value should help. Can you carry the reasoning a little further, if not to completion? Gerhard "Will Work With, Not For" Paseman, 2014.04.15
Apr
15
comment Eigenvalue of (0-1) matrix
There is also the possibility that the largest eigenvalue is less than two. To discount this possibility using the reals as the underlying field, start from the all ones vector and consider continuous perturbations to conclude the existence of a large eigenvector. I don't know if this will work over the rationals, and I don't see a combinatorial proof. Gerhard "Maybe Use Calculus As Well" Paseman, 2014.04.15
Apr
15
comment Eigenvalue of (0-1) matrix
Consider an eigenvector $a$ with maximal coordinate $a_i$. If it is associated with eigenvalue 3, that says something about some of the other coordinates. Now derive a contradiction through counting. You're welcome. Gerhard "Or Use Linear Algebra Instead" Paseman, 2014.04.15
Apr
11
comment Square-free integers not divisible by any “small” primes
I suggest going through the pain of estimating how many of those numbers have precisely c such factors for c=2,3,4. Qiaochu's heuristics above may help with that. Then you might safely conjecture how to bump it up to c=k. My guess is that the number decreases sharply when c reaches log k, which should give you nice estimates. Gerhard "Ask Me About System Design" Paseman, 2014.04.10
Apr
11
revised Placing numbers $1,2,\ldots,n^3$ in a cube so that numbers of any two adjacent unit subcube are coprime
added 804 characters in body
Apr
7
revised Placing numbers $1,2,\ldots,n^3$ in a cube so that numbers of any two adjacent unit subcube are coprime
Update on hamilton solutions for 3x3x3.
Apr
7
answered Placing numbers $1,2,\ldots,n^3$ in a cube so that numbers of any two adjacent unit subcube are coprime
Apr
3
revised Probability all inner products are zero
corrected a dumb-o, inserted another dumb-o
Apr
1
comment Probability all inner products are zero
Can you expand on the lower bound? I am expecting more like $2^{-5n/2}$. Gerhard "Likes To See Trivial Bounds" Paseman, 2014.04.01
Apr
1
comment Probability all inner products are zero
I'm getting the feeling that if $u$ has any such $v$, then $u$ or $v$ has a period length dividing $n$. Do you have any data contradicting this (e.g. a nonperiodic $u$ with at least one nonperiodic $v$ satisfying your desired relations, or a periodic $u$ with period not dividing $n$)? Gerhard "Going Through Ups And Downs" Paseman, 2014.04.01
Apr
1
comment Probability all inner products are zero
Oops. I should have said, of the $2^n$ possibilities represented by looking at the $k$ indices of the sign change, about O(k choose k/2) out of $2^k$ are allowed by the requirements. The guesstimated upper bound still looks good though. Gerhard "Returning Now To Regular Programming" Paseman, 2014.04.01
Apr
1
comment Probability all inner products are zero
Generalizing the sign change argument, it becomes clear that v_n+k is determined by the earlier v_n, giving at most $2^n$ possibilities of which all but something like O(k choose k/2) are disallowed by looking at the the indices after the k sign changes. The $2^{\epsilon + n/2}$ upper bound is looking pretty good. Gerhard "More Than Willing To Share" Paseman, 2014.04.01