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Full-Text Articles in Mathematics
Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky
Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky
CMC Faculty Publications and Research
The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current …
On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky
On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky
CMC Faculty Publications and Research
Lecture given at the Illinois Number Theory Fest, May 2007.
Stone's Representation Theorem, Ion Radu
Stone's Representation Theorem, Ion Radu
Theses Digitization Project
The thesis analyzes some aspects of the theory of distributive lattices, particularly two representation theorems: Birkhoff's representation theorem for finite distributive lattices and Stone's representation theorem for infinite distributive lattices.
Frobenius Problem And The Covering Radius Of A Lattice, Lenny Fukshansky, Sinai Robins
Frobenius Problem And The Covering Radius Of A Lattice, Lenny Fukshansky, Sinai Robins
CMC Faculty Publications and Research
Abstract. Let N ≥ 2 and let 1 < a(1) < ... < a(N) be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as Sigma(N)(i=1) a(i) x(i) where x(1),..., x(N) are non-negative integers. The condition that gcd(a(1),..., a(N)) = 1 implies that such a number exists. The general problem of determining the Frobenius number given N and a(1),..., a(N) is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.