Local Cut Points Research Articles

Big data decision tree for continuous-valued attributes based on unbalanced cut points

The decision tree is a widely used decision support model, which can quickly mine effective decision rules based on the dataset. The decision tree induction algorithm for continuous-valued attributes, based on unbalanced cut points, is efficient for mining decision rules; however, extending it to big data remains an unresolved. In this paper, two solutions are proposed to solve this problem: the first one is based on partitioning instance subsets, whereas the second one uses partitioning attribute subsets. The crucial of these two solutions is how to find the global optimal cut point from the set of local optimal cut points. For the first solution, the calculation of the Gini index of the cut points between computing nodes and the selection of the global optimal cut point by communication between these computing nodes is proposed. However, in the second solution, the division of the big data into subsets using attribute subsets in a way that all cut points of an attribute are on the same map node is proposed, the local optimal cut points can be found in this map node, then the global optimal cut point can be obtained by summarizing all local optimal cut points in the reduce node. Finally, the proposed solutions are implemented with two big data platforms, Hadoop and Spark, and compared with three related algorithms on four datasets. Experimental results show that the proposed algorithms can not only effectively solve the scalability problem, but also have lowest running time, the fastest speed and the highest efficiency under the premise of preserving the classification performance.

Connectedness and local cut points of generalized Sierpiński carpets

Connectedness and local cut points of generalized Sierpiński carpets

Local cut points and splittings of relatively hyperbolic groups

In this paper we show that the existence of a non-parabolic local cut point in the Bowditch boundary $\partial(G,\mathbb{P})$ of a relatively hyperbolic group $(G,\mathbb{P})$ implies that $G$ splits over a $2$-ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of $1$-dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over $2$-ended subgroups and no peripheral splittings. In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua $X$ and $Y$, then $Ends(X)$ is homeomorphic to $Ends(Y)$. Thus we propose an alternate definition of $Ends(G)$ which increases the class of spaces on which $G$ can act.

Boundary classification and two-ended splittings of groups with isolated flats

In this paper we provide a classification theorem for 1-dimensional boundaries of groups with isolated flats. Given a group $\Gamma$ acting geometrically on a $CAT(0)$ space $X$ with isolated flats and 1-dimensional boundary, we show that if $\Gamma$ does not split over a virtually cyclic subgroup, then $\partial X$ is homeomorphic to a circle, a Sierpinski carpet, or a Menger curve. This theorem generalizes a theorem of Kapovich-Kleiner, and resolves a question due to Kim Ruane. We also study the relationship between local cut points in $\partial X$ and splittings of $\Gamma$ over $2$-ended subgroups. In particular, we generalize a theorem of Bowditch by showing that the existence of a local point in $\partial X$ implies that $\Gamma$ splits over a $2$-ended subgroup.

Conformal dimension and canonical splittings of hyperbolic groups

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.

Central versus local adjudication of myocardial infarction in a cardiac biomarker trial

The impact of regulatory requirements, which require central adjudication for the diagnosis of acute myocardial infarction (AMI) in cardiac biomarker studies, is unclear. We determined the impact of local (at the site of subject enrollment) versus central adjudication of AMI on final diagnosis. This is a retrospective analysis of data from the Myeloperoxidase in the Diagnosis of Acute Coronary Syndromes Study, an 18-center prospective study of patients with suspected acute coronary syndromes, with enrollment from December 19, 2006, to September 20, 2007. Local adjudication of AMI was performed by a single site investigator at each center following the protocol-specified definition and according to the year 2000 definition of AMI, which based cardiac troponin (cTn) elevation on local cut points for each of the 13 different assays. After completion of the Myeloperoxidase in the Diagnosis of Acute Coronary Syndromes Study primary analysis and to evaluate a new troponin assay, a Food and Drug Administration-mandated central adjudication was performed by 3 investigators at different institutions. This adjudication used the 2007 Universal Definition of AMI, which differs by use of the manufacturer's 99th percentile cTn cut point. We describe the outcome of this process and compare it with the local adjudication. Central adjudicators were not blinded to local adjudications. For central adjudication, discrepant diagnoses were resolved by consensus. Local versus central cTn cut points differed for 6 assays. Both definitions required a rise and/or fall of cTn. Discrepant cases were reviewed by the lead author. Difficult cases were defined as having a difference between local and central adjudication, an elevated cTn with a temporal rise and fall, and a negative or absent risk stratification test. Statistics were by χ(2), κ, and logistic regression. Of 1,107 patients enrolled, 11 had indeterminate central adjudication, leaving 1,096 for analysis. In spite of high agreement across central versus local adjudicators, κ = 0.79 (95% CI [0.73, 0.85]), AMI was diagnosed more often by central adjudication, 134 (12.2%) versus 104 (9.5%), with 44 local diagnoses (4%) changed from non-AMI to AMI (n = 37) or AMI to non-AMI (n = 7) (P < .001). These 44 represented 34% (95% CI 26%-42%) of 141 cases in which either central or local adjudication was AMI. Of diagnoses changed to AMI, 3 reasons contributed approximately one-third each: the local use of a non-99th percentile cTn cutoff (32%), the possibility of human error (34%), and difficult cases (34%). Despite an acceptable κ, over a third of patients with a diagnosis of AMI were not assigned that diagnosis by both sets of adjudicators. This supports the importance of 1 standard method for diagnosis of AMI.

A Wecken theorem for n-valued maps

In [6] Schirmer (1985) established that, if φ:X⊸X is an n-valued map defined on a compact triangulable manifold of dimension at least three, then the appropriate Nielsen number, N(φ), is a sharp lower bound for the number of fixed points in the n-valued homotopy class of φ. In this note we generalize this theorem by allowing X to be any compact polyhedron without local cut points and such that no connected component is a two-manifold.

Conformal dimension and combinatorial modulus of compact metric spaces

In this Note we study the Ahlfors regular conformal dimension (dimARX) of a compact metric space X. This is a quasisymmetric numerical invariant, introduced by P. Pansu. It plays nowadays an important role in geometric group theory and in conformal dynamics. We show how to compute dimARX using a critical exponent Q associated to the combinatorial modulus. As a consequence of the equality dimARX=Q, we obtain a general criterion ensuring that the AR conformal dimension is 1. The conditions are stated in terms of local cut points of X. Finally, we give applications of these results to the boundaries of Gromov hyperbolic groups and to the Julia sets of semi-hyperbolic rational maps.

Spaces and groups with conformal dimension greater than one

We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a one-ended hyperbolic group has no local cut points, then its conformal dimension is greater than one.

Ideal boundary of 7–systolic complexes and groups

We prove that ideal boundary of a 7-systolic group is strongly hereditarily aspherical. For some class of 7-systolic groups we show their boundaries are connected and without local cut points, thus getting some results concerning splittings of those groups.

Local cut points and metric measure spaces with Ricci curvature bounded below

A local cut point is a point that disconnects its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop‐Gromov inequality. As a corollary, we obtain an upper bound for the number of ends of such a space. We also obtain some obstruction conditions for the existence of a local cut point in a metric measure space satisfying the Bishop‐Gromov inequality or the Poincare inequality. For example, the measured Gromov‐ Hausdorff limits of Riemannian manifolds with a lower Ricci curvature bound satisfy these two inequalities.

Local cutpoints and iterated clique graphs

Local cutpoints and iterated clique graphs

BOUNDARY OF CAT(-1) COXETER GROUPS

In this paper, we study the topological properties of the hyperbolic boundaries of CAT(-1) Coxeter groups of virtual cohomological dimension 2. We will show how these properties are related to combinatorial properties of the associated Coxeter graph. More precisely, we investigate the connectedness, the local connectedness and the existence problem of local cut points. In the appendix, in a joint work with Z. Sela, we will construct the JSJ decomposition of the Coxeter groups for which the corresponding Coxeter graphs are complete bipartite graphs.

Fixed point sets of deformations of polyhedra with local cut points

A locally finite simplicial complex $X$ is said to be 2-dimensionally connected if $X - \{\text {local cut points of } X \}$ is connected. Such spaces exhibit “classical” behavior in that they all admit deformations with one fixed point, and they admit fixed point free deformations if and only if the Euler characteristic is zero. A result of G.-H. Shi implies that, for non 2-dimensionally connected spaces, the fixed point sets of deformations are equivalent to the fixed point sets of certain combinatorial maps which he calls good displacements. U. K. Scholz combined Shi’s results with a theorem of P. Hall to obtain a characterization of all finite simplicial complexes which admit fixed point free deformations. In this paper we begin by explicitly capturing the combinatorial structure of a non 2-dimensionally connected polyhedron in a bipartite graph. We then apply an extended version of Hall’s theorem to this graph to get a realization theorem which gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. Scholz’s result, and a characterization of all finite simplicial complexes without fixed point free deformations that admit deformations with a single fixed point follow immediately from this realization theorem.

Local Cut-Points in Continuous Images of Compact Ordered Spaces

Local Cut-Points in Continuous Images of Compact Ordered Spaces

Local cut-points in continuous images of compact ordered spaces

We prove that if a continuum X X is continuous image of a compact ordered space and if X X is not locally separable at a point x x , then x x lies in the closure of the set of all local cut-points of X X .

Fixed points and braids. II

A basic problem in fixed point theory is to find the least number of fixed points for a homotopy class of maps. That is, given a map f : X ~ X of a connected compact polyhedron X into itself, to find M F [ f ] :=Min{ #Fix (g) lg~f :X,X} . The invariant central to this theory is the Nielson number N(f) , defined as the number of essential fixed point classes (see [1] ~)r [4]). N( f ) is always a lower bound to MF[f] and it has been a long-standing problem to prove or to disprove that N ( f ) = MFEf ]. We know [3] that if X has no local cut points and X is not a surface of negative Euler characteristic, then N ( f ) = M F [ f ] for every map f : X , X . The aim of the present paper is to generalize the result of [5] and to show

Characterizations of continua in which connected subsets are arcwise connected

The purpose of this paper is to give several characterizations of the continua in which all connected subsets are arcwise connected. The methods used are those developed by B. Knaster and K. Kuratowski, G. T. Whyburn and the author. These methods depend on Bernstein’s decomposition of a topologically complete metric space into totally imperfect sets and on Whyburn’s theory of local cutpoints. Some properties of connected sets in finitely Suslinian spaces are obtained. Two questions raised by the author are answered. Several partial results of Whyburn are obtained as corollaries of the main result.

Continua in which all connected subsets are arcwise connected

Let X be a metric continuum such that every connected subset of X is arcwise connected. Some facts concerning the distribution of local cutpoints of X are obtained. These results are used to prove that X is