856 research outputs found
A group-theoretic approach to fast matrix multiplication
We develop a new, group-theoretic approach to bounding the exponent of matrix
multiplication. There are two components to this approach: (1) identifying
groups G that admit a certain type of embedding of matrix multiplication into
the group algebra C[G], and (2) controlling the dimensions of the irreducible
representations of such groups. We present machinery and examples to support
(1), including a proof that certain families of groups of order n^(2 + o(1))
support n-by-n matrix multiplication, a necessary condition for the approach to
yield exponent 2. Although we cannot yet completely achieve both (1) and (2),
we hope that it may be possible, and we suggest potential routes to that result
using the constructions in this paper.Comment: 12 pages, 1 figure, only updates from previous version are page
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New upper bounds on sphere packings I
We develop an analogue for sphere packing of the linear programming bounds
for error-correcting codes, and use it to prove upper bounds for the density of
sphere packings, which are the best bounds known at least for dimensions 4
through 36. We conjecture that our approach can be used to solve the sphere
packing problem in dimensions 8 and 24.Comment: 26 pages, 1 figur
Algorithmic design of self-assembling structures
We study inverse statistical mechanics: how can one design a potential
function so as to produce a specified ground state? In this paper, we show that
unexpectedly simple potential functions suffice for certain symmetrical
configurations, and we apply techniques from coding and information theory to
provide mathematical proof that the ground state has been achieved. These
potential functions are required to be decreasing and convex, which rules out
the use of potential wells. Furthermore, we give an algorithm for constructing
a potential function with a desired ground state.Comment: 8 pages, 5 figure
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