pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

a cellular model for multi objects multi dimensional homotopic deformations abstract we deal in this paper with complex three dimensional scenes partitioned into several objects of any shape and with varying local dimensions we propose a consistent topological modeling of such scenes based on a cellular model we introduce a definition of simple elements for any adjacency graph based geometrical structure then we prove a local characterization of simple cells of a cellular model in the case of two objects and of any number of objects this characterization allows to define homotopic deformations of a complex scene

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

automated segmentation and classification of multispectral magnetic resonance images of brain using artificial neural networks presents a fully automated process for segmentation and classification of multispectral magnetic resonance mr images this hybrid neural network method uses a kohonen self organizing neural network for segmentation and a multilayer backpropagation neural network for classification to separate different tissue types this process uses the standard t1 t2 and pd weighted mr images acquired in clinical examinations volumetric measurements of brain structures relative to intracranial volume were calculated for an index transverse section in 14 normal subjects median age 25 years 7 male 7 female this index slice was at the level of the basal ganglia included both genu and splenium of the corpus callosum and generally showed the putamen and lateral ventricle an intraclass correlation of this automated segmentation and classification of tissues with the accepted standard of radiologist identification for the index slice in the 14 volunteers demonstrated coefficients r sub i of 0 91 0 95 and 0 98 for white matter gray matter and ventricular cerebrospinal fluid csf respectively an analysis of variance for estimates of brain parenchyma volumes in 5 volunteers imaged 5 times each demonstrated high intrasubject reproducibility with a significance of at least p 0 05 for white matter gray matter and white gray partial volumes the population variation across 14 volunteers demonstrated little deviation from the averages for gray and white matter while partial volume classes exhibited a slightly higher degree of variability this fully automated technique produces reliable and reproducible mr image segmentation and classification while eliminating intra and interobserver variability

condition k for inverse semigroups and the ideal structure of their c algebras abstract inspired by results for graph c algebras we investigate connections between the ideal structure of an inverse semigroup s and that of its tight c algebra by relating ideals in s to certain open invariant sets in the associated tight groupoid we also develop analogues of conditions l and k for inverse semigroups which are related to certain congruences on s we finish with applications to the inverse semigroups of self similar graph actions and some relevant comments on the authors earlier uniqueness theorems for inverse semigroups

the accumulation of eigenvalues in a stability problem abstract when two waves propagating in a one dimensional medium are locked together as a composite wave a natural question arises as to whether the new wave is stable an interesting and novel instability mechanism is exposed here in which a cascade of eigenvalues accumulates at a distinguished point in the unstable half plane the underlying assumption is that the transition between the two waves occurs at an unstable homogeneous steady state of the partial differential equations this causes the individual waves to have an unstable continuous spectrum but the instability of the full wave cannot be predicted from the configuration of these spectra alone

accumulations of t points in a model for solitary pulses in an excitable reaction diffusion medium we consider a family of differential equations that describes traveling waves in a reaction diffusion equation modeling oxidation of carbon oxide on a platinum surface near the onset of spatio temporal chaos the organizing bifurcation for the bifurcation structure with small carbon oxide pressures turns out to be a codimension 3 bifurcation involving a homoclinic orbit to an equilibrium undergoing a transcritical bifurcation we show how infinitely many t point bifurcations of multi loop heteroclinic cycles occur in the unfolding

pseudogroups and their etale groupoids abstract a pseudogroup is a complete infinitely distributive inverse monoid such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces the goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory c algebras and aperiodic tilings our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids as a corollary to this duality we deduce a non commutative version of stone duality involving what we call boolean inverse semigroups and boolean etale groupoids as well as a generalization of this duality to distributive inverse semigroups non commutative stone duality has important applications in the theory of c algebras it is the basis for the construction of cuntz and cuntz krieger algebras and in the case of the cuntz algebras it can also be used to construct the thompson groups we then define coverages on inverse semigroups and the resulting presentations of pseudogroups as applications we show that paterson s universal groupoid is an example of a booleanization and reconcile exel s recent work on the theory of tight maps with the work of the second author

higher dimensional generalizations of the thompson groups abstract motivated by the goal of generalizing the cuntz krieger algebras and making heavy use of the pioneering work of robertson and steger kumjian and pask generalized the notion of a directed graph to what they termed a higher rank graph the c algebras constructed from such higher rank graphs have proved to be highly interesting higher rank graphs are in fact a class of cancellative categories and so the one vertex higher rank graphs are therefore a class of cancellative monoids in this paper we call them k monoids finite direct products of free monoids belong to this family but there are many examples not of this form it is well known that the classical thompson groups arise from the free monoids and it is easy to show that brin s higher dimensional thompson groups arise from finite direct products of free monoids in this paper we generalize these constructions and show how to construct a family of new groups with simple commutator subgroups from arbitrary aperiodic k monoids thereby subsuming all the above examples although we give simple direct constructions of our groups a key feature of our paper is that we also show that they arise as topological full groups of the usual etale groupoids associated with k monoids in this way our groups are naturally associated with c algebras the question of the mutual isomorphisms amongst this family of groups appears to be a delicate one and is not handled here

global bifurcations close to symmetry abstract heteroclinic cycles involving two saddle foci where the saddle foci share both invariant manifolds occur persistently in some symmetric differential equations on the 3 dimensional sphere we analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1 dimensional connection the saddle foci have the same chirality when part of the symmetry is broken the 2 dimensional invariant manifolds intersect transversely creating a heteroclinic network of bykov cycles we show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics there are n pulse heteroclinic tangencies trajectories that follow the original cycle n times around before they arrive at the other node each n pulse heteroclinic tangency is accumulated by a sequence of n 1 pulse ones this coexists with the suspension of horseshoes defined on an infinite set of disjoint strips where the first return map is hyperbolic we also show how as the system approaches full symmetry the suspended horseshoes are destroyed creating regions with infinitely many attracting periodic solutions

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

spiralling dynamics near heteroclinic networks abstract there are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically we construct explicitly a two parameter family of vector fields on the three dimensional sphere s 3 whose flow has a spiralling attractor containing the following two hyperbolic equilibria heteroclinic trajectories connecting them transversely and a non trivial hyperbolic invariant and transitive set the spiralling set unfolds a heteroclinic network between two symmetric saddle foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols coexisting with newhouse phenomena the vector field is the restriction to s 3 of a polynomial vector field in r 4 in this article we also identify global bifurcations that induce chaotic dynamics of different types

computation of the mid sagittal plane in 3 d brain images we present a new method to automatically compute reorient and recenter the mid sagittal plane in anatomical and functional three dimensional 3 d brain images this iterative approach is composed of two steps at first given an initial guess of the mid sagittal plane generally the central plane of the image grid the computation of local similarity measures between the two sides of the head allows to identify homologous anatomical structures or functional areas by way of a block matching procedure the output is a set of point to point correspondences the centers of homologous blocks subsequently we define the mid sagittal plane as the one best superposing the points on one side and their counterparts on the other side by reflective symmetry practically the computation of the parameters characterizing the plane is performed by a least trimmed squares estimation then the estimated plane is aligned with the center of the image grid and the whole process is iterated until convergence the robust estimation technique we use allows normal or abnormal asymmetrical structures or areas to be treated as outliers and the plane to be mainly computed from the underlying gross symmetry of the brain the algorithm is fast and accurate even for strongly tilted heads and even in presence of high acquisition noise and bias field as shown on a large set of synthetic data the algorithm has also been visually evaluated on a large set of real magnetic resonance mr images we present a few results on isotropic as well as anisotropic anatomical mr and computed tomography and functional single photon emission computed tomography and positron emission tomography real images for normal and pathological subjects

general non commutative locally compact locally hausdorff stone duality abstract we extend the classical stone duality between zero dimensional compact hausdorff spaces and boolean algebras specifically we simultaneously remove the zero dimensionality restriction and extend to etale groupoids obtaining a duality with an elementary class of inverse semigroups

graph inverse semigroups their characterization and completion abstract graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of c algebras in this paper we characterize such semigroups and show how they may be completed under suitable conditions to form what we call the cuntz krieger semigroup of the graph this semigroup is proved to be the ample semigroup of a topological groupoid associated with the graph and the semigroup analogue of the leavitt path algebra of the graph

homoclinic phenomena in laser models abstract a description of the principal bifurcations which lead to the appearance of the lorenz attractor is given for the 3d normal form for codimension 3 bifurcations of equilibria and periodic orbits in systems with symmetry we pay special attention to two bifurcation points corresponding to the formation of a homoclinic butterfly of a saddle with unit saddle index and to a homoclinic butterfly with zero separatrix value

antiextensive connected operators for image and sequence processing this paper deals with a class of morphological operators called connected operators these operators filter the signal by merging its flat zones as a result they do not create any new contours and are very attractive for filtering tasks where the contour information has to be preserved this paper shows that connected operators work implicitly on a structured representation of the image made of flat zones the max tree is proposed as a suitable and efficient structure to deal with the processing steps involved in antiextensive connected operators a formal definition of the various processing steps involved in the operator is proposed and as a result several lines of generalization are developed first the notion of connectivity and its definition are analyzed several modifications of the traditional approach are presented they lead to connected operators that are able to deal with texture they also allow the definition of connected operators with less leakage than the classical ones second a set of simplification criteria are proposed and discussed they lead to simplicity entropy and motion oriented operators the problem of using a nonincreasing criterion is analyzed its solution is formulated as an optimization problem that can be very efficiently solved by a viterbi 1979 algorithm finally several implementation issues are discussed showing that these operators can be very efficiently implemented

topological equivalence of tilings we introduce a notion of equivalence on tilings which is formulated in terms of their local structure we compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic

the non transverse shil nikov hopf bifurcation uncoupling of homoclinic orbits and homoclinic tangencies abstract it is known that in a neighbourhood of a codimension two shil nikov hopf bifurcation primary periodic orbits lie on a single wiggly curve in period parameter space with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit in contrast it has recently been shown by hirschberg and laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit this paper analyses the codimension three bifurcation caused by a non transverse i e degenerately parametrically unfolded shil nikov hopf bifurcation which contains nearby dynamics akin to both degeneracies two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle focus homoclinic orbits to equilibria or the uncoupling of this locus from the locus of hopf bifurcations we undertake a heuristic analysis of the unfolding showing that in both cases it contains codimension two non transversal homoclinic orbits to equilibria and non transversal homoclinic tangencies to periodic orbits unfolding the former non transverse orbit is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas asymptotic expressions are given for the accumulation of two types of isola forming bifurcations the implications of z 2 equivariance on the unfolding is discussed finally numerical evidence is presented for both upward and downward pointing non transverse shil nikov hopf bifurcations occuring in a model of an autonomous non linear electronic circuit both with respect to a z 2 equivariant and a z 2 non equivariant equilibrium numerical computation of curves of periodic orbits homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

analytical search for bifurcation surfaces in parameter space the method of resultants can be used to compute hopf bifurcations in ode systems we discuss this method from the applicant s point of view furthermore we show by theory and examples that the method may be extended to cover other bifurcation situations as well among them are the real hopf situation which plays an important role in the transition to shil nikov chaos as well as some higher codimension bifurcations such as takens bogdanov gavrilov guckenheimer and double hopf bifurcations the method yields an analytical test function that can be solved analytically or by computer algebra systems in contrast to common analytical techniques based on eigenvalue computation which can only be applied to systems of size n 4 the method is applicable for systems of intermediate size n 10 we illustrate the power of the method by discussing examples from different disciplines of science a lorenz like oscillator two coupled oscillators and a five species food chain

quantales and fell bundles abstract we study fell bundles on groupoids from the viewpoint of quantale theory given any saturated upper semicontinuous fell bundle e g on an etale groupoid g with g 0 locally compact hausdorff equipped with a suitable completion c algebra a of its convolution algebra we obtain a map of involutive quantales p max a g where max a consists of the closed linear subspaces of a and g is the topology of g we study various properties of p which mimick to various degrees those of open maps of topological spaces these are closely related to properties of g and a such as g being hausdorff principal or topological principal or being a line bundle under suitable conditions which include g being hausdorff but without requiring saturation of the fell bundle a is an algebra of sections of the bundle if and only if it is the reduced c algebra c r g e we also prove that max a is stably gelfand this implies the existence of a pseudogroup i b and of an etale groupoid b associated canonically to any sub c algebra b a we study a correspondence between fell bundles and sub c algebras based on these constructions and compare it to the construction of weyl groupoids from cartan subalgebras

etale groupoids and their quantales abstract we establish close and previously unknown relations between quantales and groupoids in particular to each etale groupoid either localic or topological there is associated a unital involutive quantale we obtain a bijective correspondence between localic etale groupoids and their quantales which are given a rather simple characterization and here are called inverse quantal frames we show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids and as a consequence we obtain a non functorial correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids this generalization is entirely algebraic and it is valid in an arbitrary topos as a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen and an analogue of this holds for topological groupoids in practice we are provided with new tools for constructing localic and topological etale groupoids as well as inverse semigroups for instance via presentations of quantales by generators and relations the characterization of inverse quantal frames is to a large extent based on a new quantale operation here called a support whose properties are thoroughly investigated and which may be of independent interest

the local structure of tilings and their integer group of coinvariants the local structure of a tiling is described in terms of a multiplicative structure on its pattern classes the groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined this group furnishes part of the k0 group of the groupoid c algebra for tilings which reduce to decorations of the group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and v primitive this yields in particular the set of possible gap labels predicted by k theory for schrodinger operators describing the particle motion in such a tiling

segmentation registration and measurement of shape variation via image object shape a model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives metrics are described that compute an object representation s prior probability of local geometry by reflecting variabilities in the net s node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation a paradigm for image analysis of deforming such a model to optimize a posteriori probability is described and this paradigm is shown to be usable as a uniform approach for object definition object based registration between images of the same or different imaging modalities and measurement of shape variation of an abnormal anatomical object compared with a normal anatomical object examples of applications of these methods in radiotherapy surgery and psychiatry are given

diagonal preserving isomorphisms of etale groupoid algebras abstract work of jean renault shows that for topologically principal etale groupoids a diagonal preserving isomorphism of reduced c algebras yields an isomorphism of groupoids several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses in this paper we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids our approach has the additional feature that we only need to impose conditions on one of the two groupoids applications are given to leavitt path algebras

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

the bifurcations of separatrix contours and chaos abstract bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points are studied and cases of the appearance of nontrivial hyperbolic sets are pointed out

varieties of boolean inverse semigroups in an earlier work the author observed that boolean inverse semigroups with semigroup homomorphisms preserving finite orthogonal joins form a congruence permutable variety of algebras called biases we give a full description of varieties of biases in terms of varieties of groups 1 every free bias is residually finite in particular the word problem for free biases is decidable 2 every proper variety of biases contains a largest finite symmetric inverse semigroup and it is generated by its members that are monoids of generalized rook matrices over groups with zero 3 there is an order preserving one to one correspondence between proper varieties of biases and certain finite sequences of varieties of groups descending in a strong sense defined in terms of wreath products by finite symmetric groups

the many groupoids of a stably gelfand quantale abstract we study the projections of an arbitrary stably gelfand quantale q and show that each projection determines a pseudogroup s q and a corresponding localic etale groupoid g together with a map of involutive quantales p q l s o g as an application we obtain a simplified axiomatization of inverse quantal frames quantales of etale groupoids whereby such a quantale is shown to be the same as a unital stably gelfand quantal frame q whose partial units cover q

inverse semigroups associated to subshifts abstract the dynamics of a one sided subshift x can be modeled by a set of partially defined bijections from this data we define an inverse semigroup s x and show that it has many interesting properties we prove that the carlsen matsumoto c algebra o x associated to x is canonically isomorphic to exel s tight c algebra of s x as one consequence we obtain that o x can be written as a partial crossed product of a commutative c algebra by a countable group

a perspective on non commutative frame theory abstract this paper extends the fundamental results of frame theory to a non commutative setting where the role of locales is taken over by etale localic categories this involves ideas from quantale theory and from semigroup theory specifically ehresmann semigroups restriction semigroups and inverse semigroups we prove several main results to start with we establish a duality between the category of complete restriction monoids and the category of etale localic categories the relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames this result builds on the work of pedro resende on the connection between pseudogroups and etale localic groupoids but in the process we both generalize and simplify for example we do not require involutions and in addition we render his result functorial a wider class of quantales called multiplicative ehresmann quantal frames is put into a correspondence with those localic categories where the multiplication structure map is semiopen and all the other structure maps are open we also project down to topological spaces and as a result extend the classical adjunction between locales and topological spaces to an adjunction between etale localic categories and etale topological categories in fact varying morphisms we obtain several adjunctions just as in the commutative case we restrict these adjunctions to spatial sober and coherent spectral equivalences the classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups consequently we deduce several dualities between distributive restriction semigroups and spectral etale topological categories we also specialize these dualities for the setting where the topological categories are cancellative or are groupoids our approach thus links unifies and extends the approaches taken in the work by lawson and lenz and by resende

matching 3 d anatomical surfaces with non rigid deformations using octree splines presents a new method for determining the minimal nonrigid deformation between two 3d surfaces such as those which describe anatomical structures in 3d medical images although the authors match surfaces they represent the deformation as a volumetric transformation their method performs a least squares minimization of the distance between the two surfaces of interest to quickly and accurately compute distances between points on the two surfaces the authors use a precomputed distance map represented using an octree spline whose resolution increases near the surface to quickly and robustly compute the deformation the authors use a second octree spline to model the deformation function the coarsest level of the deformation encodes the global e g affine transformation between the two surfaces while finer levels encode smooth local displacements which bring the two surfaces into closer registration the authors present experimental results on both synthetic and real 3d surfaces

topologically controlled segmentation of 3d magnetic resonance images of the head by using morphological operators abstract this paper proposes a new data driven segmentation technique of 3d t1 weighted magnetic resonance scans of human head this technique serves to the construction of individual head models several structures of the head are extracted the morphology oriented approach combined with an extensive use of topological constraints provides a robust and automatic method requiring minimum user intervention this new approach is suitable to applications where the topology is one of the main constraints the originality of the approach lies in the satisfaction of such constraints and in an effort towards robustness

subgroups of the group of homeomorphisms of the cantor space and a duality between a class of inverse monoids and a class of hausdorff etale groupoids abstract under non commutative stone duality there is a correspondence between second countable hausdorff etale groupoids which have a cantor space of identities and what we call tarski inverse monoids that is countable boolean inverse monoids with semilattices of idempotents which are countable and atomless tarski inverse monoids are therefore the algebraic counterparts of the etale groupoids studied by matui and provide a natural setting for many of his calculations under this duality we prove that natural properties of the etale groupoid correspond to natural algebraic properties of the tarski inverse monoid effective groupoids correspond to fundamental tarski inverse monoids and minimal groupoids correspond to 0 simplifying tarski inverse monoids particularly interesting are the principal groupoids which correspond to tarski inverse monoids where every element is a finite join of infinitesimals and idempotents here an infinitesimal is simply a non zero element with square zero the groups of units of fundamental tarski inverse monoids generalize the finite symmetric groups and include amongst their number the thompson groups g n 1 as well as the groups of units of af inverse monoids krieger s ample groups being examples

on models with non rough poincare homoclinic curves abstract the possibility of an a priori complete description of finite parameter models including systems with structurally unstable poincare homoclinic curves is studied the main result reported here is that systems having a countable set of moduli of equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the newhouse regions of non stability we discuss the question of correctly setting a problem for the analysis of models of such type