Decompositions of Fundamental Groups of Closed Surfaces into Free Constructions

Let T be a closed surface. It is proven that any decomposition of ₁(T,x) into an amalgamated product (or, more generally, into the fundamental group of a finite graph of groups) with f.g. edge group(s) is almost geometric. A problem of H. Zieschang is solved and the edge rigidity property is investigated.     

Short Loop Decompositions of Surfaces and the Geometry of Jacobians

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions.     

Extremal Decompositions of Riemann Surfaces and Quasiconformal Mappings of Special Form

It is shown that the extremal decomposition of a finite Riemann surface into a system of doubly connected domains may be associated with a family of quasiconformal mappings , which are similar to the Teichmüller mappings. In the case , this construction allows us to prove that the extremal value of the functional in the indicated problem on the extremal decomposition is a pluriharmonic function of the coordinates of the distinguished points on . Bibliography: 5 titles.     

Extremal Decompositions of Riemann Surfaces and Quasiconformal Mappings of Special Form. II

Let be a finite Riemann surface. For a quadratic differential on associated with a certain problem on extremal decomposition of into n domains, a parametric family of quasiconformal mappings , is defined. These mappings map the domains of the extremal decomposition of onto the domains of the extremal decomposition of ._K This allows one to study the functional dependence of the problem on the parameters. Bibliography: 6 titles.     

Random Walk on Surfaces with Hyperbolic Cusps

We consider the operator associated with a random walk on finite volume surfaces with hyperbolic cusps. We study the spectral gap (upper and lower bound) associated with this operator and deduce some rate of convergence of the iterated kernel towards its stationary distribution.     

Decompositions of rings

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 137–139, January, 1995.     

Decompositions of comodules

In this paper we study the relations among the different decomposition theories of comodules.     

Decompositions of degenerations

We develop a method to refine a given degeneration of modules. This enables us to give a new proof of the equivalence of the partial orders ≤_deg and ≤_ext in the case that the algebra Λ has a directed Auslander–Reiten quiver. Received: 11 November 1999     

Decompositions of semigroups

Communicated by Gerard J. Lallement     

Decompositions of signed-graphic matroids

We give a decomposition theorem for signed graphs whose frame matroids are binary and a decomposition theorem for signed graphs whose frame matroids are quaternary.     

External Paradoxical Decompositions

Ore's condition states that a cancellative semigroup S which has common right multiples embeds into a group G such that certain properties are satisfied by S and G. We show that G is nonamenable if and only if the semigroup S^-1 is G-paradoxical with respect to right multiplication by elements of S. We explore certain properties of this decomposition of S^-1.     

Sparsity-certifying Graph Decompositions

We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. Ileana Streinu; Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author.     

Decompositions of binomial ideals

We present Binomials, a package for the computer algebra system Macaulay 2, which specializes well-known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that significant speedup of computations like primary decomposition is possible. While central parts of the implemented algorithms go back to a paper of Eisenbud and Sturmfels, we also discuss a new algorithm for computing the minimal primes of a binomial ideal. All decompositions make significant use of combinatorial structure found in binomial ideals, and to demonstrate the power of this approach we show how Binomials was used to compute primary decompositions of commuting birth and death ideals of Evans et al., yielding a counterexample for their conjectures.     

Decompositions of determinantal varieties

In this paper, we study the irreducible decompositions of determinantal varieties of matrices given by rank conditions on upper left submatrices. Using the concept of essential rank function and the Ehresmann partial order on the set of all simple matrices, we design an algorithm to write a determinantal variety as a union of its irreducible components. This solves a problem raised by B. Sturmfels.