On the upper bound of the number of limit cycles obtained by the second order averaging method II

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Geometric tools to determine the hyperbolicity of limit cycles

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Algebraic extensions in free groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical fieldt heoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

The mean dehn functions of Abelian groups

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Traverso's isogeny conjecture for p-divisible groups

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Multiplicity of limit cycles and analytic m-solutions for planar differential systems

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Minimal lenght elements of Thompson's groups F(p)

We describe a method for determining the minimal length of elements in the generalized Thompson's groups F(p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.

Renormalization and α-limit set for expanding Lorenz maps

We establish a one-to-one correspondence between the renormalizations and proper totally invariant closed sets (i.e., α-limit sets) of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. We describe the minimal renormalization by constructing the minimal totally invariant closed set, so that we can define the renormalization operator. Using consecutive renormalizations, we obtain complete topological characteriza- tion of α-limit sets and nonwandering set decomposition. For piecewise linear Lorenz map with slopes ≥ 1, we show that each renormalization is periodic and every proper α-limit set is countable.

On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

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Contraction groups in complete Kac-Moody groups

Let G be an abstract Kac-Moody group over a finite field and G the closure of the image of G in the automorphism group of its positive building. We show that if the Dynkin diagram associated to G is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in G which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)

Parabolic and quasiparabolic subgroups of free partially commutative groups

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions.

Discriminating groups: a comprehensive overview

Discriminating groups were introduced by G.Baumslag, A.Myasnikov and V.Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. However they have taken on a life of their own and have been an object of a considerable amount of study. In this paper we survey the large array results concerning the class of discriminating groups that have been developed over the past decade.

The equation xpyq=zr and groups that act freely on Λ-trees

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Octopus Standard Automated Perimetry, Pulsar Perimetry, Moorfields Motion Displacement Test and Heidelberg Retinal Tomography in Glaucoma Detection - A Comparison of Diagnostic Accuracy

Purpose: To investigate the accuracy of 4 clinical instruments in the detection of glaucomatous damage. Methods: 102 eyes of 55 test subjects (Age mean = 66.5yrs, range = [39; 89]) underwent Heidelberg Retinal Tomography (HRTIII), (disc area<2.43); and standard automated perimetry (SAP) using Octopus (Dynamic); Pulsar (TOP); and Moorfields Motion Displacement Test (MDT) (ESTA strategy). Eyes were separated into three groups 1) Healthy (H): IOP<21mmHg and healthy discs (clinical examination), 39 subjects, 78 eyes; 2) Glaucoma suspect (GS): Suspicious discs (clinical examination), 12 subjects, 15 eyes; 3) Glaucoma (G): progressive structural or functional loss, 14 subjects, 20 eyes. Clinical diagnostic precision was examined using the cut-off associated with the p<5% normative limit of MD (Octopus/Pulsar), PTD (MDT) and MRA (HRT) analysis. The sensitivity, specificity and accuracy were calculated for each instrument. Results: See table Conclusions: Despite the advantage of defining glaucoma suspects using clinical optic disc examination, the HRT did not yield significantly higher accuracy than functional measures. HRT, MDT and Octopus SAP yielded higher accuracy than Pulsar perimetry, although results did not reach statistical significance. Further studies are required to investigate the structure-function correlations between these instruments.

Orbit decidability and the conjugacy problem for some extensions of groups

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