34 research outputs found

### Computing spectral sequences

In this paper, a set of programs enhancing the Kenzo system is presented.
Kenzo is a Common Lisp program designed for computing in Algebraic Topology, in
particular it allows the user to calculate homology and homotopy groups of
complicated spaces. The new programs presented here entirely compute Serre and
Eilenberg-Moore spectral sequences, in particular the groups and differential
maps for arbitrary r. They also determine when the spectral sequence has
converged and describe the filtration of the target homology groups induced by
the spectral sequence

### Homotopy types of stabilizers and orbits of Morse functions on surfaces

Let $M$ be a smooth compact surface, orientable or not, with boundary or
without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$
the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule
$h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in
C^{\infty}(M,P)$.
Let $f:M \to P$ be a Morse function and $O(f)$ be the orbit of $f$ under this
action. We prove that $\pi_k O(f)=\pi_k M$ for $k\geq 3$, and $\pi_2 O(f)=0$
except for few cases. In particular, $O(f)$ is aspherical, provided so is $M$.
Moreover, $\pi_1 O(f)$ is an extension of a finitely generated free abelian
group with a (finite) subgroup of the group of automorphisms of the Reeb graph
of $f$.
We also give a complete proof of the fact that the orbit $O(f)$ is tame
Frechet submanifold of $C^{\infty}(M,P)$ of finite codimension, and that the
projection $Diff(M) \to O(f)$ is a principal locally trivial $S(f)$-fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that
the orbits of a finite codimension of tame action of tame Lie group on tame
Frechet manifold is a tame Frechet manifold itsel

### Reusing integer homology information of binary digital images

In this paper, algorithms for computing integer (co)homology of a simplicial complex of any dimension are designed, extending the work done in [1,2,3]. For doing this, the homology of the object is encoded in an algebraic-topological format (that we call AM-model). Moreover, in the case of 3D binary digital images, having as input AM-models for the images I and J, we design fast algorithms for computing the integer homology of I âˆªJ, I âˆ©J and I âˆ–J

### Local structure of the set of steady-state solutions to the 2D incompressible Euler equations

It is well known that the incompressible Euler equations can be formulated in
a very geometric language. The geometric structures provide very valuable
insights into the properties of the solutions. Analogies with the
finite-dimensional model of geodesics on a Lie group with left-invariant metric
can be very instructive, but it is often difficult to prove analogues of
finite-dimensional results in the infinite-dimensional setting of Euler's
equations. In this paper we establish a result in this direction in the simple
case of steady-state solutions in two dimensions, under some non-degeneracy
assumptions. In particular, we establish, in a non-degenerate situation, a
local one-to-one correspondence between steady-states and co-adjoint orbits.Comment: 81 page

### Using membrane computing for obtaining homology groups of binary 2D digital images

Membrane Computing is a new paradigm inspired from cellular communication. Until now, P systems have been used in research areas like modeling chemical process, several ecosystems, etc. In this paper, we apply P systems to Computational Topology within the context of the Digital Image. We work with a variant of P systems called tissue-like P systems to calculate in a general maximally parallel manner the homology groups of 2D images. In fact, homology computation for binary pixel-based 2D digital images can be reduced to connected component labeling of white and black regions. Finally, we use a software called Tissue Simulator to show with some examples how these systems wor

### Computing with Locally Effective Matrices

In this work, we start from the naive notion of integer infinite matrix (i.e., the functions of the set Ã— = {f: Ã— }). Then, several undecidability results are established, leading to a convenient data structure for effective machine computations. We call this data structure a locally effective matrix. We study when (and how) the standard matrix calculus (Ker and CoKer computations) can be extended to the infinite case. We find again several undecidability barriers. When these limitations are overcome, we describe effective procedures for computing in the locally effective case. Finally, the role played by these data structures in the development of real symbolic computation systems for algebraic topology (based on the effective homology notion) is illustrated

### Computing with locally effective matrices

In this work, we start from the naive notion of integer infinite matrix (i.e., the functions of the set Z NÃ—N ={f: N Ã— N â†’ Z}). Then, several undecidability results are established, leading to a convenient data structure for effective machine computations. We call this data structure a locally effective matrix. We study when (and how) the standard matrix calculus (Ker and CoKer computations) can be extended to the infinite case. We find again several undecidability barriers. When these limitations are overcome, we describe effective procedures for computing in the locally effective case. Finally, the role played by these data structures in the development of real symbolic computation systems for algebraic topology (based on the effective homology notion) is illustrated