Gerhard Paseman
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Unregistered User
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Apr 4 |
awarded | ● Nice Answer |
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Feb 5 |
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Asking about a quasicommutative semigroup added 563 characters in body |
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Feb 5 |
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Asking about a quasicommutative semigroup Perhaps the problem is more basic. I will edit after the light dawns and fills the corners of my cranium. Gerhard "Quantifier Application Is Not Commutative" Paseman, 2013.02.05 |
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Feb 5 |
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Asking about a quasicommutative semigroup OK. I'm in trouble now. Please help. ab= b^ra=a^rb^r =ba since rth powers commute. Or do they? Gerhard "Shouldn't Be Using Martin's Definition?" Paseman, 2013.02.05 |
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Feb 5 |
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Asking about a quasicommutative semigroup I speak hastily, of course. All of the above is for classes of semigroups which share the same r. So I not only assume that r is the same for all a and b (minor quibble, as for a finite semigroup I can take R something like product or max of all r), but also every semigroup in the class uses the same r (not so minor). Gerhard "Now Return To Regular Programming" Paseman, 2013.02.05 |
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Feb 5 |
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Asking about a quasicommutative semigroup After spending two more minutes thinking, I see we can take w to be not only squarefree but of length 1. Thus either I am very confused, or we have that quasicommutative semigroups form a locally finite variety. Gerhard "Has Been Very Confused Before" Paseman, 2013.02.05 |
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Feb 5 |
answered | Asking about a quasicommutative semigroup |
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Feb 5 |
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Mathematical techniques to reduce the amount of storage memory Quid, his specification is not explicit enough: some of the requirements have to be inferred from the phrase "Big Data". Either n will be large or k will be large or both. In either case, I see it as compressing an array of k binary strings of n bits. For ease of discussion, I assume a skewed distribution of array values, so that the most common entries are stored using d bit codes for each common value, with d much smaller than n. The problem is a common one in CS and industry, with a variety of solutions. Gerhard "Ask Me About System Design" Paseman, 2013.02.05 |
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Feb 5 |
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Mathematical techniques to reduce the amount of storage memory Look up dictionary compression. Especially in the case that k is much greater than 2^n, the idea is to encode common bit vectors by a shorter string that uses fewer than n bits, and use longer strings for combinations that are less rare. This works primarily if the distribution of patterns is far from uniform. For a scheme that works for all possible distributions, bit-vectors are liikely the way to go. Gerhard "Ask Me About Saving Memory" Paseman, 2013.02.05 |
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Feb 5 |
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Finding a vertex equidistant from two given vertices in a digraph Aaron's suggestion makes me think that there is a reduction from the Frobenius coin problem to this one. However, the reduction I am thinking of is exponential; it may be that doing a reduction in the other direction will yield a polynomial time solution (to borrow Aaron's example, polynomial in 6, 10 and 15 and not polynomial in their logs). I don't see such a reduction being more clever than Aaron's adjacency matrix suggestion. Gerhard "Reduce, Recycle, And Reuse Mathematics" Paseman, 2013.02.04 |
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Feb 4 |
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Ancient method to study Archimedean spiral With just a modest amount of being charitable Emil, I see implicit in the question something similar to "What is a weak first-order theory or other logical theory that doesn't need calculus or modern machinery to do what I ask about and that is close to something Archimedes could have used?" . A logical perspective on formal systems that model reasoning in ancient times might help with this and similar questions of interest. Gerhard "That's How I See It" Paseman, 2013.02.04 |
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Feb 4 |
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Full-rank linearly independent matrices Consider the cycle q=(1 2 3 ...n) on the set of columns of an order n matrix over F_2 with n > 1. Let Q=q applied to the order n identity matrix, so the main diagonal is shifted "out of the way". In addition to the n(n-1) matrices I + E_ij for i distinct from j, take also the n matrices Q + E_ii. I think this or a slight modification to resolve parity issues should work as a basis in characteristic 2. Gerhard "Ask Me About System Design" Paseman, 2013.02.03 |
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Feb 3 |
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Suggestions for good notation I can see that being handy when one needs to call out some of the indices. For your example from Bourbaki, I would sooner use X decorated with a hat or overbar to indicate a tuple. Gerhard "That May Just Be Me" Paseman, 2013.02.03 |
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Feb 3 |
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Is the empty graph a tree? I remember attending a MSRI conference in a previous millenium on universal algebra and category theory. I was wondering if a religious battle would break out over the notion of an empty algebra. Fortunately no blood was shed at that conference over the issue. This post reminds me of those years. Tom, I'm afraid I dealt more with nonempty algebras than potentially empty relational structures, so I am not entirely convinced by your comment. Not that you should worry. Gerhard "Pronounces Tom Devoid Of Blame" Paseman, 2013.02.02 |
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Feb 2 |
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Choosing a base where a given digit of a given number appears the most times If you aren't picky, choosing base 1 or some irrational base will likely work. I don't think you should invent a term like "oneier" though. Gerhard "Inventing Words Is Sorta Fun" Paseman, 2013.02.02 |
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Feb 2 |
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Which popular games are the most mathematical? I think my point is that some view games mathematically, for the goal of practicing mathematics. While I understand and often sympathize with such an endeavour, I remind you that some games are for pure social entertainment, and mathematical analysis is often counter to that goal. I would consider a semiotic approach to Charades analysis, and I am not clear what goals your suggested approach are trying to reach. Gerhard "Likes Board Over Party Games" Paseman, 2013.02.01 |
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Feb 1 |
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Full-rank linearly independent matrices If n > 1, any order n matrix is the sum of exactly two invertible matrices, even if the field has characteristic two. So I would expect an invertible basis for characteristic 2 also. Gerhard "Ask Me About Binary Matrices" Paseman, 2013.02.01 |
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Feb 1 |
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Is the empty graph a tree? Trees have roots and leaves, and when they are big enough they have branches as well. The one vertex graph is a not big tree, and the empty graph is not a tree, in my view. (The empty graph could be soil, or a dog, or a pink elephant, depending on the metaphor. Unless you need an additive identity though, the empty graph is not a tree.) Gerhard "It Is What You Need" Paseman, 2013.02.01 |
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Feb 1 |
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Number of possible circuits with N NOR gates and M inputs The above is still rough, but (as in a MathOverflow comment about Graham's number that has disappeared from me) it is larger than the actual bound, so still is an upper bound. One thing to note is that R is not N, since I disinguish combinatorial (early) gates from result (final) gates whose inputs depend on outputs of gates that are not result gates. An interesting problem is on limiting FF, which should be feasible since all F outputs are used, which limits the arrangements of 2F inputs to the feedback gates substantially. Gerhard "Always Looking For Smaller Numbers" Paseman, 2013.02.01 |
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Feb 1 |
answered | Number of possible circuits with N NOR gates and M inputs |
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Jan 31 |
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Number of possible circuits with N NOR gates and M inputs Unfortunately, the entries to Boolean Functions don't quite cut it, as some of the designs produce oscillators. Gerhard "Circuits Is A Better Term" Paseman, 2013.01.31 |
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Jan 31 |
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Representations with Triangular Numbers The space of numbers m which are themselves the sum of two triangular numbers may be thick enough to answer the question, as for those numbers the product is seen to be less than m. This suggests a smart and greedy approach where the target is a sum of certain pairs of triangular numbers. Gerhard "Sometimes Sharing And Not Greedy" Paseman, 2013.01.31 |
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Jan 31 |
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Number of possible circuits with N NOR gates and M inputs Also, by looking at permuting the n outputs, you might be able to remove a factor of close to n factorial from the bound to remove some redundancy. Gerhard "Ask Me About System Design" Paseman, 2013.01.30 |
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Jan 31 |
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Number of possible circuits with N NOR gates and M inputs A weak bound is ((m+n+1) choose 2)^n, as each of the gates has at most (m+n+1) choose 2 possibilities for each of the inputs. If you do a brute force enumeration for small n, you may be able to determine the dependence on the number of inputs pretty easily. Certainly it will be easy once you have found the values for m at most 2n. Gerhard "Ask Me About Gate Arrays" Paseman, 2013.01.30 |
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Jan 31 |
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Which popular games are the most mathematical? How would you start an analysis of the party game Charades? Gerhard "Or Worse Yet, Of Pictionary" Paseman, 2013.01.30 |
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Jan 29 |
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Edge-coloring of the complete graph without any rainbow paths I don't understand. For k=2 and the suggested coloring on K_4, I get a rainbow path of 3 0 2 1, and I get similar paths for larger k. What did I do wrong? Gerhard "Did I Flip A Sign?" Paseman, 2013.01.28 |
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Jan 28 |
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Numbers of a certain form not expressible as squares Also, while I presently do not see how to improve on Chein's argument, I still think that something simpler and possibly less complete is wanted. Gerhard "AskMe About System Design" Paseman, 2013.01.28 |
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Jan 28 |
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Numbers of a certain form not expressible as squares I commented above (and then mistakenly deleted) that the specific result desired by the poster was realized through mod 3 considerations. I suspect that the poster is even more interested in when elementary considerations provide quick answers. For example, knowing when 2 is not a qth power mod a covers the cases mentioned and many more. Gerhard "Looks For Really Simple Answers" Paseman, 2013.01.28 |
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Jan 28 |
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Counting square-free numbers smoothly I'm with km. Indeed, for any set of integers n of positive density, I see the desired sum over that set as infinite. Gerhard "Ask Me About Unbounded Confusion" Paseman, 2013.01.28 |
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Jan 28 |
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what is this small 3 element quasigroup & what is it used for I appreciate the action taken to undelete the question. meta.mathoverflow.net/discussion/1522/… a thread to request the reopening of this question. I encourage the original poster to edit the question as suggested (at the moment I cannot perform the edit). It is my hope to supply information found in Berman's classification that answers what I see as a reference request. Gerhard "Ask Me About System Design" Paseman, 2013.01.27 |
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Jan 26 |
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Need tight lower bound for independence number of order 10 graph Towards your goal of furthering new theory, you might present ten such minimal graphs and ask what else they may have in common; that might spur the development of new and better lower bounds. Gerhard "Ask Me About System Design" Paseman, 2013.01.25 |
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Jan 26 |
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Need tight lower bound for independence number of order 10 graph Try numerical order. I found such a set of 4 independent points. For such graphs, I find using the triangles helps. Gerhard "Ask Me About System Design" Paseman, 2013.01.25 |
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Jan 25 |
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Graphs with circulant distance matrices Other cyclic like graphs work, e.g. the complete graph, as well as complements of such graphs. Gerhard "Ask Me About System Design" Paseman, 2013.01.25 |
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Jan 25 |
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Sum of binomial coefficient For many values of epsilon less than 1/2, d will be such that n choose d is close to epsilon2^n, or one more. For many epsilon between 1/2 and 1, you can use an analogous value based on n-d. It is an interesting exercise to find how close to 1/2 that finding such a value becomes challenging. Perhaps others will give you references to other MathOverflow questions where this sum has been discussed. Gerhard "Too Lazy To Search Now" Paseman, 2013.01.24 |
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Jan 24 |
revised |
sorting two paired lists of real numbers to minimize consecutive absolute differences added 495 characters in body |
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Jan 23 |
answered | sorting two paired lists of real numbers to minimize consecutive absolute differences |
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Jan 23 |
awarded | ● Yearling |
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Jan 17 |
answered | Maximum distance within a subset of permutations |
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Jan 17 |
answered | Hamiltonian cycles in power-graphs |
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Jan 15 |
answered | What is the maximal sparsity of a matrix? |
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Jan 4 |
answered | power of adjacency matrix |
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Dec 29 |
answered | Math for a cake |
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Dec 25 |
answered | Smallest square to wrap a cylinder |
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Dec 20 |
answered | Similarity measure between 2 bi-partite graph. |
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Dec 19 |
answered | Excellent mathematical explanations |
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Dec 15 |
answered | Positive results coming from paradoxes |
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Dec 12 |
answered | How should one look at the set of compatible ring structures on a given group? |
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Dec 8 |
answered | Sieve of Erathostenes: removing consecutive items |
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Dec 2 |
awarded | ● Yearling |
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Dec 2 |
awarded | ● Yearling |
