3,396 research outputs found
Logspace computations in graph products
We consider three important and well-studied algorithmic problems in group
theory: the word, geodesic, and conjugacy problem. We show transfer results
from individual groups to graph products. We concentrate on logspace complexity
because the challenge is actually in small complexity classes, only. The most
difficult transfer result is for the conjugacy problem. We have a general
result for graph products, but even in the special case of a graph group the
result is new. Graph groups are closely linked to the theory of Mazurkiewicz
traces which form an algebraic model for concurrent processes. Our proofs are
combinatorial and based on well-known concepts in trace theory. We also use
rewriting techniques over traces. For the group-theoretical part we apply
Bass-Serre theory. But as we need explicit formulae and as we design concrete
algorithms all our group-theoretical calculations are completely explicit and
accessible to non-specialists
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
On the enumeration of closures and environments with an application to random generation
Environments and closures are two of the main ingredients of evaluation in
lambda-calculus. A closure is a pair consisting of a lambda-term and an
environment, whereas an environment is a list of lambda-terms assigned to free
variables. In this paper we investigate some dynamic aspects of evaluation in
lambda-calculus considering the quantitative, combinatorial properties of
environments and closures. Focusing on two classes of environments and
closures, namely the so-called plain and closed ones, we consider the problem
of their asymptotic counting and effective random generation. We provide an
asymptotic approximation of the number of both plain environments and closures
of size . Using the associated generating functions, we construct effective
samplers for both classes of combinatorial structures. Finally, we discuss the
related problem of asymptotic counting and random generation of closed
environemnts and closures
Graph products of spheres, associative graded algebras and Hilbert series
Given a finite, simple, vertex-weighted graph, we construct a graded
associative (non-commutative) algebra, whose generators correspond to vertices
and whose ideal of relations has generators that are graded commutators
corresponding to edges. We show that the Hilbert series of this algebra is the
inverse of the clique polynomial of the graph. Using this result it easy to
recognize if the ideal is inert, from which strong results on the algebra
follow. Non-commutative Grobner bases play an important role in our proof.
There is an interesting application to toric topology. This algebra arises
naturally from a partial product of spheres, which is a special case of a
generalized moment-angle complex. We apply our result to the loop-space
homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more
citations, to appear in Mathematische Zeitschrif
VC-saturated set systems
The well-known Sauer lemma states that a family of VC-dimension at most has size at most
. We obtain both random and explicit constructions to
prove that the corresponding saturation number, i.e., the size of the smallest
maximal family with VC-dimension , is at most , and thus is
independent of
Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
We derive the analog of the large Gross-Taylor holomorphic string
expansion for the refinement of -deformed Yang-Mills theory on a
compact oriented Riemann surface. The derivation combines Schur-Weyl duality
for quantum groups with the Etingof-Kirillov theory of generalized quantum
characters which are related to Macdonald polynomials. In the unrefined limit
we reproduce the chiral expansion of -deformed Yang-Mills theory derived by
de Haro, Ramgoolam and Torrielli. In the classical limit , the expansion
defines a new -deformation of Hurwitz theory wherein the refined
partition function is a generating function for certain parameterized Euler
characters, which reduce in the unrefined limit to the orbifold Euler
characteristics of Hurwitz spaces of holomorphic maps. We discuss the
geometrical meaning of our expansions in relation to quantum spectral curves
and -ensembles of matrix models arising in refined topological string
theory.Comment: 45 pages; v2: References adde
Computing Adapted Bases for Conformal Automorphism Groups of Riemann Surfaces
The concept of an adapted homology basis for a prime order conformal
automorphism of a compact Riemann surface extends to arbitrary finite groups of
conformal automorphisms. Here we compute some examples of adapted homology
bases for some groups of automorphisms. The method is to begin by apply the
Schreier-Reidemeister rewriting process along with the Schreier-Reidemeister
Theorem and then to eliminate generators and relations until there is one
single large defining relation for the fundamental group in which every
generator and its inverse occurs. We are then able to compute the action of the
group on the homology image of these generators in the first homology group.
The matrix of the action is in a simple form. This has applications to the
representation variety.Comment: Typos and spelling error fixed; 17 pages; to appear AMS Conn Math;
Proc. Linkopoing Conference, 201
- …