1,834 research outputs found
A note on the equivariant formal group law of the equivariant complex cobordism ring
For a finite abelian group , we compute the -equivariant formal group
law corresponding to the -equivariant complex cobordism spectrum with its
canonical complex orientation.Comment: 11 pages, v
Intersections of multiplicative translates of 3-adic Cantor sets
Motivated by a question of Erd\H{o}s, this paper considers questions
concerning the discrete dynamical system on the 3-adic integers given by
multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers
whose expansions use only the digits 0 and 1. The exception set is the set of
3-adic integers whose forward orbits under this action intersects the 3-adic
Cantor set infinitely many times. It has been shown that this set has Hausdorff
dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these
sets leads to study of intersections of multiplicative translates of Cantor
sets by powers of 2. More generally, this paper studies the structure of finite
intersections of general multiplicative translates of the 3-adic Cantor set by
integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3-adic
integers whose 3-adic expansions have one-sided symbolic dynamics given by a
finite automaton. As a consequence, the Hausdorff dimension of such a set is
always of the form log(\beta) for an algebraic integer \beta. This paper gives
a method to determine the automaton for given data (M_1, ..., M_n).
Experimental results indicate that the Hausdorff dimension of such sets depends
in a very complicated way on the integers M_1,...,M_n.Comment: v1, 31 pages, 6 figure
p-adic path set fractals and arithmetic
This paper considers a class C(Z_p) of closed sets of the p-adic integers
obtained by graph-directed constructions analogous to those of Mauldin and
Williams over the real numbers. These sets are characterized as collections of
those p-adic integers whose p-adic expansions are describeed by paths in the
graph of a finite automaton issuing from a distinguished initial vertex. This
paper shows that this class of sets is closed under the arithmetic operations
of addition and multiplication by p-integral rational numbers. In addition the
Minkowski sum (under p-adic addition) of two set in the class is shown to also
belong to this class. These results represent purely p-adic phenomena in that
analogous closure properties do not hold over the real numbers. We also show
the existence of computable formulas for the Hausdorff dimensions of such sets.Comment: v1 24 pages; v2 added to title, 28 pages; v3, 30 pages, added
concluding section, v.4, incorporate changes requested by reviewe
Path sets in one-sided symbolic dynamics
Path sets are spaces of one-sided infinite symbol sequences associated to
pointed graphs (G_v_0), which are edge-labeled directed graphs G with a
distinguished vertex v_0. Such sets arise naturally as address labels in
geometric fractal constructions and in other contexts. The resulting set of
symbol sequences need not be closed under the one-sided shift. this paper
establishes basic properties of the structure and symbolic dynamics of path
sets, and shows they are a strict generalization of one-sided sofic shifts.Comment: 16 pages, 6 figures; v2, 22pages, 6 figures; title change, adds a new
Theorem 1.5, and a second Appendix, v3, 21 pages, revisions to exposition; v4
revised introduction; v5, 22 pages, changed title, revised introductio
Distributed Approximation Algorithms for Weighted Shortest Paths
A distributed network is modeled by a graph having nodes (processors) and
diameter . We study the time complexity of approximating {\em weighted}
(undirected) shortest paths on distributed networks with a {\em
bandwidth restriction} on edges (the standard synchronous \congest model). The
question whether approximation algorithms help speed up the shortest paths
(more precisely distance computation) was raised since at least 2004 by Elkin
(SIGACT News 2004). The unweighted case of this problem is well-understood
while its weighted counterpart is fundamental problem in the area of
distributed approximation algorithms and remains widely open. We present new
algorithms for computing both single-source shortest paths (\sssp) and
all-pairs shortest paths (\apsp) in the weighted case.
Our main result is an algorithm for \sssp. Previous results are the classic
-time Bellman-Ford algorithm and an -time
-approximation algorithm, for any integer
, which follows from the result of Lenzen and Patt-Shamir (STOC 2013).
(Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use
to hide the O(\poly\log n) term.) We present an -time -approximation algorithm for \sssp. This
algorithm is {\em sublinear-time} as long as is sublinear, thus yielding a
sublinear-time algorithm with almost optimal solution. When is small, our
running time matches the lower bound of by Das Sarma
et al. (SICOMP 2012), which holds even when , up to a
\poly\log n factor.Comment: Full version of STOC 201
Gyroscopic polynomials
Gyroscopic alignment of a fluid occurs when flow structures align with the
rotation axis. This often gives rise to highly spatially anisotropic columnar
structures that in combination with complex domain boundaries pose challenges
for efficient numerical discretizations and computations. We define gyroscopic
polynomials to be three-dimensional polynomials expressed in a coordinate
system that conforms to rotational alignment. We remap the original domain with
radius-dependent boundaries onto a right cylindrical or annular domain to
create the computational domain in this coordinate system. We find the volume
element expressed in gyroscopic coordinates leads naturally to a hierarchy of
orthonormal bases. We build the bases out of Jacobi polynomials in the vertical
and generalized Jacobi polynomials in the radial. Because these coordinates
explicitly conform to flow structures found in rapidly rotating systems the
bases represent fields with a relatively small number of modes. We develop the
operator structure for one-dimensional semi-classical orthogonal polynomials as
a building block for differential operators in the full three-dimensional
cylindrical and annular domains. The differential operators of generalized
Jacobi polynomials generate a sparse linear system for discretization of
differential operators acting on the gyroscopic bases. This enables efficient
simulation of systems with strong gyroscopic alignment
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