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

Social Networking Data

__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)


6h
comment improving known bounds for Pierce expansions; cash prize
Now that I spent minutes in the lab, I've spent seconds reading the 1996 pdf which reproduces the results in the comments above, except that max S_9 is 20160, and my table for min S_r measures something slightly different. I imagine the $2 connection has already been explored. Gerhard "Decades Late, Two Dollars Short" Paseman, 2014.04.23
6h
comment improving known bounds for Pierce expansions; cash prize
Also, for each a in S_r, I suspect the smallest b for which P(a,b)=r satisfies a < 2b < a + O(log^2 a), but I have not verified this yet. Gerhard "Just Adding My Two Cents" Paseman, 2014.04.23
6h
comment improving known bounds for Pierce expansions; cash prize
Min S_r is 1,3,5,13,11,19,35,47,53,95, 103,179,251,299,503,743,1019,1319,1439,2939, 3359,3959,6619,5387,5879, a mostly monotonic sequence for r from 1 to 26, with several entries of high integer complexity. I would set this problem in front of those working on integer complexity and see what they think. (Of course, these are results from a quickly written and possibly error prone program. The line of attack should still be worth pursuing.) Gerhard "Is It Coincidence, Or Murder?!?" Paseman, 2014.04.23
7h
comment improving known bounds for Pierce expansions; cash prize
Here's something that might be worth at least $2: I'm seeing some vague similarities between this problem and the one's complexity problem. Let S_r be the set of those a for which P(a,b) <= r for all b < a, and with equality for at least 1 b. The sequence of S_r bears some resemblance to sequence of T_r, the set of numbers with one complexity = r (see arxiv:1404.2183 by de Reyna and van de Lune for a recent treatment). In particular, max S_r = 2,6,24,72,240,720,2880,6720,9240 for r from 1 through 9. Gerhard "But Wait! There is More!" Paseman, 2014.04.23
2d
comment Advice on choosing an area of specialization
Do both. Pick something that will be a successful thesis, and show your capacity as a researcher, and work on your passion at the same time. If in the middle of your efforts you find your passion consuming/replacing your "job", get help in restoring balance. Alternatively, if in the middle you hit on something in your passion that allows you to replace your thesis topic, ask your advisors about that possibility. Gerhard "Passion Equals Job Implies Satisfaction" Paseman, 2014.04.21
2d
comment Sets of cardinalities of bases without choice
If you are able to talk about the subclass of script BS which arise from well-ordered bases, you should be able to show that subclass is closed under intersection. You might then explore what weakenings (larger subclasses) you have in which you can show things like closed under finite intersection. I recommend going through a universal algebra text and seeing which basic constructions are choiceless or require a minimum amount (e.g. Galois connections, congruence lattices). You may end up with something CS-theory people are doing. Gerhard "Giving Noah S's HS Theorem" Paseman, 2014.04.21
2d
comment Distance between two networks
As a step toward your goal, consider all maps f: between A and B. Devise a measure on this space that determines how far a map is from an isomorphism, say count of tuples abar such that R(abar) and not R(f(abar)) for a relation R the network has (number of nodes of degree d as an example, but you may want several differing R.) If you are looking for a distance, now see if your measure will give you the additional properties between different spaces. Gerhard "Ask Me About System Design" Paseman, 2014.04.21
2d
comment Fixed point problem with a monotone vector as a fixed point?
If you don't have a condition on $F$ that "favors some coordinates over others", then for any $F$ that might have your property, there should be a function $ G = \pi^{-1}\circ F \circ \pi $ where $\pi$ is effectively a permutation matrix that shuffles the coordinates of the input vector, and may induce a different fixed point without the desired order. Gerhard "Feels Like Something Is Missing" Paseman, 2014.04.21
2d
comment Integral straight-line embeddings of planar graphs
Heiko, not Heikod. (If I could think of 5 more characters to change, I would make the edit myself.) Gerhard "Ask Me About Jacobsthal's Function" Paseman, 2014.04.21
2d
comment Integer sequences such that each term forms k-consecutive composite integers
Possible? Yes. Enumerate prime gaps and focus on gaps large enough for the designated k. Efficient? There will be uncountably many (consider numbers near n! to start), and the distribution of gaps is not well understood, so there will not be a nice expression beyond "look for $a_n$ somewhere near the beginning of the $m$th prime gap of length larger than $k$, for $m$ sometimes at least $n$." If you need something more specific (e.g. describing many such sequences by a polynomial expression) you might reframe your question. Gerhard "Ask Me About Jacobsthal's Function" Paseman, 2014.04.21
Apr
17
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
Apr
17
revised Advice for number theory library
added 21 characters in body
Apr
17
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
Apr
17
answered Advice for number theory library
Apr
17
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
Apr
17
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
Apr
15
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
Apr
15
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