283 research outputs found
Virtual knot groups and almost classical knots
We define a group-valued invariant of virtual knots and relate it to various
other group-valued invariants of virtual knots, including the extended group of
Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A
virtual knot is called almost classical if it admits a diagram with an
Alexander numbering, and in that case we show that the group factors as a free
product of the usual knot group and Z. We establish a similar formula for mod p
almost classical knots, and we use these results to derive obstructions to a
virtual knot K being mod p almost classical. Viewed as knots in thickened
surfaces, almost classical knots correspond to those that are homologically
trivial. We show they admit Seifert surfaces and relate their Alexander
invariants to the homology of the associated infinite cyclic cover. We prove
the first Alexander ideal is principal, recovering a result first proved by
Nakamura et al. using different methods. The resulting Alexander polynomial is
shown to satisfy a skein relation, and its degree gives a lower bound for the
Seifert genus. We tabulate almost classical knots up to 6 crossings and
determine their Alexander polynomials and virtual genus.Comment: 44 page
SAWdoubler: a program for counting self-avoiding walks
This article presents SAWdoubler, a package for counting the total number
Z(N) of self-avoiding walks (SAWs) on a regular lattice by the length-doubling
method, of which the basic concept has been published previously by us. We
discuss an algorithm for the creation of all SAWs of length N, efficient
storage of these SAWs in a tree data structure, and an algorithm for the
computation of correction terms to the count Z(2N) for SAWs of double length,
removing all combinations of two intersecting single-length SAWs.
We present an efficient numbering of the lattice sites that enables
exploitation of symmetry and leads to a smaller tree data structure; this
numbering is by increasing Euclidean distance from the origin of the lattice.
Furthermore, we show how the computation can be parallelised by distributing
the iterations of the main loop of the algorithm over the cores of a multicore
architecture. Experimental results on the 3D cubic lattice demonstrate that
Z(28) can be computed on a dual-core PC in only 1 hour and 40 minutes, with a
speedup of 1.56 compared to the single-core computation and with a gain by
using symmetry of a factor of 26. We present results for memory use and show
how the computation is made to fit in 4 Gbyte RAM. It is easy to extend the
SAWdoubler software to other lattices; it is publicly available under the GNU
LGPL license.Comment: 29 pages, 3 figure
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States
We introduce a unifying and generalizing framework for complex and detailed
balanced steady states in chemical reaction network theory. To this end, we
generalize the graph commonly used to represent a reaction network.
Specifically, we introduce a graph, called a reaction graph, that has one edge
for each reaction but potentially multiple nodes for each complex. A special
class of steady states, called node balanced steady states, is naturally
associated with such a reaction graph. We show that complex and detailed
balanced steady states are special cases of node balanced steady states by
choosing appropriate reaction graphs. Further, we show that node balanced
steady states have properties analogous to complex balanced steady states, such
as uniqueness and asymptotical stability in each stoichiometric compatibility
class. Moreover, we associate an integer, called the deficiency, to a reaction
graph that gives the number of independent relations in the reaction rate
constants that need to be satisfied for a positive node balanced steady state
to exist.
The set of reaction graphs (modulo isomorphism) is equipped with a partial
order that has the complex balanced reaction graph as minimal element. We
relate this order to the deficiency and to the set of reaction rate constants
for which a positive node balanced steady state exists
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
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