20,506 research outputs found
Distinguishing subgroups of the rationals by their Ramsey properties
A system of linear equations with integer coefficients is partition regular
over a subset S of the reals if, whenever S\{0} is finitely coloured, there is
a solution to the system contained in one colour class. It has been known for
some time that there is an infinite system of linear equations that is
partition regular over R but not over Q, and it was recently shown (answering a
long-standing open question) that one can also distinguish Q from Z in this
way.
Our aim is to show that the transition from Z to Q is not sharp: there is an
infinite chain of subgroups of Q, each of which has a system that is partition
regular over it but not over its predecessors. We actually prove something
stronger: our main result is that if R and S are subrings of Q with R not
contained in S, then there is a system that is partition regular over R but not
over S. This implies, for example, that the chain above may be taken to be
uncountable.Comment: 14 page
Finite Type Invariants of w-Knotted Objects II: Tangles, Foams and the Kashiwara-Vergne Problem
This is the second in a series of papers dedicated to studying w-knots, and
more generally, w-knotted objects (w-braids, w-tangles, etc.). These are
classes of knotted objects that are wider but weaker than their "usual"
counterparts. To get (say) w-knots from usual knots (or u-knots), one has to
allow non-planar "virtual" knot diagrams, hence enlarging the the base set of
knots. But then one imposes a new relation beyond the ordinary collection of
Reidemeister moves, called the "overcrossings commute" relation, making
w-knotted objects a bit weaker once again. Satoh studied several classes of
w-knotted objects (under the name "weakly-virtual") and has shown them to be
closely related to certain classes of knotted surfaces in R4. In this article
we study finite type invariants of w-tangles and w-trivalent graphs (also
referred to as w-tangled foams). Much as the spaces A of chord diagrams for
ordinary knotted objects are related to metrized Lie algebras, the spaces Aw of
"arrow diagrams" for w-knotted objects are related to not-necessarily-metrized
Lie algebras. Many questions concerning w-knotted objects turn out to be
equivalent to questions about Lie algebras. Most notably we find that a
homomorphic universal finite type invariant of w-foams is essentially the same
as a solution of the Kashiwara-Vergne conjecture and much of the
Alekseev-Torossian work on Drinfel'd associators and Kashiwara-Vergne can be
re-interpreted as a study of w-foams.Comment: 57 pages. Improvements to the exposition following a referee repor
Limitations of Algebraic Approaches to Graph Isomorphism Testing
We investigate the power of graph isomorphism algorithms based on algebraic
reasoning techniques like Gr\"obner basis computation. The idea of these
algorithms is to encode two graphs into a system of equations that are
satisfiable if and only if if the graphs are isomorphic, and then to (try to)
decide satisfiability of the system using, for example, the Gr\"obner basis
algorithm. In some cases this can be done in polynomial time, in particular, if
the equations admit a bounded degree refutation in an algebraic proof systems
such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on
the polynomial calculus degree over all fields of characteristic different from
2 and also linear lower bounds for the degree of Positivstellensatz calculus
derivations.
We compare this approach to recently studied linear and semidefinite
programming approaches to isomorphism testing, which are known to be related to
the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the
power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system
that lies between degree-k Nullstellensatz and degree-k polynomial calculus
Partition regularity without the columns property
A finite or infinite matrix A with rational entries is called partition
regular if whenever the natural numbers are finitely coloured there is a
monochromatic vector x with Ax=0. Many of the classical theorems of Ramsey
Theory may naturally be interpreted as assertions that particular matrices are
partition regular. In the finite case, Rado proved that a matrix is partition
regular if and only it satisfies a computable condition known as the columns
property. The first requirement of the columns property is that some set of
columns sums to zero.
In the infinite case, much less is known. There are many examples of matrices
with the columns property that are not partition regular, but until now all
known examples of partition regular matrices did have the columns property. Our
main aim in this paper is to show that, perhaps surprisingly, there are
infinite partition regular matrices without the columns property --- in fact,
having no set of columns summing to zero.
We also make a conjecture that if a partition regular matrix (say with
integer coefficients) has bounded row sums then it must have the columns
property, and prove a first step towards this.Comment: 13 page
Quantum supergroups and topological invariants of three - manifolds
The Reshetikhin - Turaeve approach to topological invariants of three -
manifolds is generalized to quantum supergroups. A general method for
constructing three - manifold invariants is developed, which requires only the
study of the eigenvalues of certain central elements of the quantum supergroup
in irreducible representations. To illustrate how the method works,
at odd roots of unity is studied in detail, and the
corresponding topological invariants are obtained.Comment: 22 page
On 3-dimensional lattice walks confined to the positive octant
Many recent papers deal with the enumeration of 2-dimensional walks with
prescribed steps confined to the positive quadrant. The classification is now
complete for walks with steps in : the generating function is
D-finite if and only if a certain group associated with the step set is finite.
We explore in this paper the analogous problem for 3-dimensional walks
confined to the positive octant. The first difficulty is their number: there
are 11074225 non-trivial and non-equivalent step sets in
(instead of 79 in the quadrant case). We focus on the 35548 that have at most
six steps.
We apply to them a combined approach, first experimental and then rigorous.
On the experimental side, we try to guess differential equations. We also try
to determine if the associated group is finite. The largest finite groups that
we find have order 48 -- the larger ones have order at least 200 and we believe
them to be infinite. No differential equation has been detected in those cases.
On the rigorous side, we apply three main techniques to prove D-finiteness.
The algebraic kernel method, applied earlier to quadrant walks, works in many
cases. Certain, more challenging, cases turn out to have a special Hadamard
structure, which allows us to solve them via a reduction to problems of smaller
dimension. Finally, for two special cases, we had to resort to computer algebra
proofs. We prove with these techniques all the guessed differential equations.
This leaves us with exactly 19 very intriguing step sets for which the group
is finite, but the nature of the generating function still unclear.Comment: Final version, to appear in Annals of Combinatorics. 36 page
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