[math-fun] Erdos conjecture on arithmetic progressions
Erdos conjectured that if the reciprocal sum of a sequence of positive integers diverges, then it contains k-arithmetic progressions for any k. It is known* that this implies the existence of constants N_k such that if the sum of the reciprocals of a sequence is greater than N_k, the sequence contains a k-arithmetic progression. Of all subsets of positive integers not containing a 3-arithmetic progression, what's the highest reciprocal sum you can find? The greedy method gets over 3.0078. * Gerver 1977, Brown & Freedman 1987 in a stronger form Charles Greathouse Analyst/Programmer Case Western Reserve University
Can anyone please point me to any article or book that addresses the question of what are the maximal finite subgroups of GL(n,Z) ? What I'm ultimately interested in is this: Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ? (Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.) E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions. Thanks, Dan
I'd recommend reading Sphere Packings, Lattices and Groups, by J. H. Conway and N. J. A. Sloane. That lists all of the known interesting lattices in small numbers of dimensions. Sincerely, Adam P. Goucher
Can anyone please point me to any article or book that addresses the question of what are the maximal finite subgroups of GL(n,Z) ?
What I'm ultimately interested in is this:
Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ?
(Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.)
E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions.
Thanks,
Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
A better reference is - *146. **Low-Dimensional Lattices II: Subgroups of GL(n, Z)<http://neilsloane.com/doc/Me146.pdf> *, J. H. Conway and N. J. A. Sloane, *Proc. Royal Soc. London, Series A*, 419 (1988), pp. 29-68. available from my home page (see below), item 146. On Sat, Jul 28, 2012 at 3:58 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
I'd recommend reading Sphere Packings, Lattices and Groups, by J. H. Conway and N. J. A. Sloane. That lists all of the known interesting lattices in small numbers of dimensions.
Sincerely,
Adam P. Goucher
Can anyone please point me to any article or book that addresses the
question of what are the maximal finite subgroups of GL(n,Z) ?
What I'm ultimately interested in is this:
Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ?
(Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.)
E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions.
Thanks,
Dan ______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
Stupid questions raised by rapid perusal of Conway & Sloane (1988): Dimensions 10,12,14,15,16 etc. are not mentioned. What makes these cases harder or less interesting than dimensions 11,13,17,19,23 ? Dimensions 11,19 are included in the early pages; but later sections omit discussion of their lattices. Why is this? Fred Lunnon On 7/28/12, Neil Sloane <njasloane@gmail.com> wrote:
A better reference is
- *146. **Low-Dimensional Lattices II: Subgroups of GL(n, Z)<http://neilsloane.com/doc/Me146.pdf> *, J. H. Conway and N. J. A. Sloane, *Proc. Royal Soc. London, Series A*, 419 (1988), pp. 29-68.
available from my home page (see below), item 146.
On Sat, Jul 28, 2012 at 3:58 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
I'd recommend reading Sphere Packings, Lattices and Groups, by J. H. Conway and N. J. A. Sloane. That lists all of the known interesting lattices in small numbers of dimensions.
Sincerely,
Adam P. Goucher
Can anyone please point me to any article or book that addresses the
question of what are the maximal finite subgroups of GL(n,Z) ?
What I'm ultimately interested in is this:
Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ?
(Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.)
E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions.
Thanks,
Dan ______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thanks very much, Neil, for the pointer to your paper. Now I will need to figure out which if any of these "maximally symmetric" lattices (besides A*_2 and A*_3) have Voronoi tessellations with vertex figures that are simplices. Any further pointers to where this information might be found will be greatly appreciated. -Dan On 2012-07-28, at 12:39 PM, Dan Asimov wrote:
Can anyone please point me to any article or book that addresses the question of what are the maximal finite subgroups of GL(n,Z) ?
What I'm ultimately interested in is this:
Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ?
(Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.)
E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions.
There are more related papers on http://neilsloane.com/doc/pub.html Three titles contain "Voronoi": 82. Voronoi Regions of Lattices, Second Moments of Polytopes, and Quantization, J. H. Conway and N. J. A. Sloane, IEEE Trans. Information Theory, IT-28 (1982), pp. 211-226, A revised version appears as Chapter 21 of ``Sphere Packings, Lattices and Groups'' by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988 Reprinted in ``Vector Quantization'', H. Abut, Ed., IEEE Press, NY, 1990, pp. 118-133.. 108. On the Voronoi Regions of Certain Lattices, J. H. Conway and N. J. A. Sloane, SIAM J. Algebraic Discrete Methods, 5 (1984), pp. 294-305. 169. Low-Dimensional Lattices VI: Voronoi Reduction of Three-Dimensional Lattices, J. H. Conway and N. J. A. Sloane, Proc. Royal Soc. London, Series A, 436 (1992), pp. 55-68. [Errata: 1) In the proof of Theorem 2: N(v-2w) < N(w) should read N(v-2w) < N(v); 2) Equation (1) is missing a minus sign before p_ij; 3) in equation (12), v'_13 has the wrong sign.] * Dan Asimov <dasimov@earthlink.net> [Jul 30. 2012 15:25]:
Thanks very much, Neil, for the pointer to your paper.
Now I will need to figure out which if any of these "maximally symmetric" lattices (besides A*_2 and A*_3) have Voronoi tessellations with vertex figures that are simplices.
Any further pointers to where this information might be found will be greatly appreciated.
-Dan
On 2012-07-28, at 12:39 PM, Dan Asimov wrote:
Can anyone please point me to any article or book that addresses the question of what are the maximal finite subgroups of GL(n,Z) ?
What I'm ultimately interested in is this:
Which n-dimensional lattices L (in R^n) have a maximal automorphism group Aut(L) -- maximal as a subgroup of O(n) ?
(Aut(L) here is the group of isometries of R^n that fix the origin and take L onto L.)
E.g., in 2D these are the square and triangular lattices. In 3D I think these are the cubic, bcc and fcc lattices. I don't know the answer for higher dimensions.
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participants (6)
-
Adam P. Goucher -
Charles Greathouse -
Dan Asimov -
Fred lunnon -
Joerg Arndt -
Neil Sloane