The Experts below are selected from a list of 46839 Experts worldwide ranked by ideXlab platform
Christopher Tuffley - One of the best experts on this subject based on the ideXlab platform.
-
Finite Subset spaces of closed surfaces
arXiv: Geometric Topology, 2003Co-Authors: Christopher TuffleyAbstract:The kth Finite Subset space of a topological space X is the space exp_k X of non-empty Finite Subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We show that the Finite Subset spaces of a connected 2-complex admit "lexicographic cell structures" based on the lexicographic order on I^2 and use these to study the Finite Subset spaces of closed surfaces. We completely calculate the rational homology of the Finite Subset spaces of the two-sphere, and determine the top integral homology groups of exp_k Sigma for each k and closed surface Sigma. In addition, we use Mayer-Vietoris arguments and the ring structure of H^*(Sym^k Sigma) to calculate the integer cohomology groups of the third Finite Subset space of Sigma closed and orientable.
-
connectivity of Finite Subset spaces of cell complexes
arXiv: Geometric Topology, 2003Co-Authors: Christopher TuffleyAbstract:The kth Finite Subset space of a topological space X is the space exp_k X of non-empty Subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the Finite Subset spaces of connected graphs we show that the kth Finite Subset space of a connected cell complex is (k-2)-connected, and (k-1)-connected if in addition the underlying space is simply connected. We expect exp_k X to be (k+m-2)-connected if X is an m-connected cell complex, and reduce proving this to the problem of proving it for Finite wedges of (m+1)-spheres. Our results complement a theorem due to Handel that for path-connected Hausdorff X the map on pi_i induced by the inclusion exp_k X --> exp_{2k+1} X is zero for all k and i.
-
Finite Subset spaces of s1
Algebraic & Geometric Topology, 2002Co-Authors: Christopher TuffleyAbstract:Given a topological space X denote by exp k X the space of non-empty Subsets of X of size at most k, topologised as a quotient of X k . This space may be regarded as a union over 1 ≤ l < k of configuration spaces of l distinct unordered points in X. In the special case X = S 1 we show that: (1) exp k S 1 has the homotopy type of an odd dimensional sphere of dimension k or k - 1; (2) the natural inclusion of exp 2 k - 1 S 1 ≃ S 2 k - 1 into exp 2 k S 1 ≃ S 2 k - 1 is multiplication by two on homology; (3) the complement exp k S 1 \ exp k - 2 S 1 of the codimension two strata in exp k S 1 has the homotopy type of a (k - 1, k)-torus knot complement; and (4) the degree of an induced map exp k f: exp k S 1 → exp k S 1 is (deg f) [ ( k + 1 ) / 2 ] for f: S 1 → S 1 . The first three results generalise known facts that exp 2 S 1 is a Mobius strip with boundary exp 1 S 1 , and that exp 3 S 1 is the three-sphere with exp 1 S 1 inside it forming a trefoil knot.
-
Finite Subset spaces of graphs and punctured surfaces
arXiv: Geometric Topology, 2002Co-Authors: Christopher TuffleyAbstract:The kth Finite Subset space of a topological space X is the space exp_k X of non-empty Finite Subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We calculate the homology of the Finite Subset spaces of a connected graph Gamma, and study the maps (exp_k phi)_* induced by a map phi:Gamma -> Gamma' between two such graphs. By homotopy functoriality the results apply to punctured surfaces also. The braid group B_n may be regarded as the mapping class group of an n-punctured disc D_n, and as such it acts on H_*(exp_k D_n). We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at most floor((n-1)/2).
-
Finite Subset spaces of s 1
arXiv: Geometric Topology, 2002Co-Authors: Christopher TuffleyAbstract:Given a topological space X denote by exp_k(X) the space of non-empty Subsets of X of size at most k, topologised as a quotient of X^k. This space may be regarded as a union over 0 exp_k(S^1) is (deg f)^[(k+1)/2] for f: S^1-->S^1. The first three results generalise known facts that exp_2(S^1) is a Moebius strip with boundary exp_1(S^1), and that exp_3(S^1) is the three-sphere with exp_1(S^1) inside it forming a trefoil knot.
Oded Maler - One of the best experts on this subject based on the ideXlab platform.
-
preemptive job shop scheduling using stopwatch automata
Tools and Algorithms for Construction and Analysis of Systems, 2002Co-Authors: Yasmina Abdeddaïm, Oded MalerAbstract:In this paper we show how the problem of job-shop scheduling where the jobs are preemptible can be modeled naturally as a shortest path problem defined on an extension of timed automata, namely stopwatch automata where some of the clocks might be freezed at certain states. Although general verification problems on stopwatch automata are known to be undecidable, we show that due to particular properties of optimal schedules, the shortest path in the automaton belongs to a Finite Subset of the set of acyclic paths and hence the problem is solvable. We present several algorithms and heuristics for finding the shortest paths in such automata and test their implementation on numerous benchmark examples.
A A Tuzhilin - One of the best experts on this subject based on the ideXlab platform.
-
fermat steiner problem in the metric space of compact sets endowed with hausdorff distance
Journal of Geometry, 2017Co-Authors: Alexandr Olegovich Ivanov, Alexandr Tropin, A A TuzhilinAbstract:Fermat–Steiner problem consists in finding all points in a metric space Y such that the sum of distances from each of them to the points from some fixed Finite Subset A of Y is minimal. Such points are sometimes referred as geometric medians of A. This problem is investigated for the metric space \(Y=H(X)\) of compact Subsets of a metric space X, endowed with the Hausdorff distance. For the case of a proper metric space X a description of all compacts \(K\in H(X)\) which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct a surprising example of a quite symmetric regular triangle in \(H(\mathbb {R}^2)\), such that all its shortest trees have no “natural” symmetry.
-
fermat steiner problem in the metric space of compact sets endowed with hausdorff distance
arXiv: Metric Geometry, 2016Co-Authors: Alexandr Olegovich Ivanov, Alexandr Tropin, A A TuzhilinAbstract:The Fermat-Steiner problem consists in finding all points in a metric space $Y$ such that the sum of distances from each of them to the points from some fixed Finite Subset of $Y$ is minimal. This problem is investigated for the metric space $Y=H(X)$ of compact Subsets of a metric space $X$, endowed with the Hausdorff distance. For the case of a proper metric space $X$ a description of all compacts $K\in H(X)$ which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct an example of a regular triangle in $H(R^2)$, such that all its shortest trees have no "natural" symmetry.
P S Thiagarajan - One of the best experts on this subject based on the ideXlab platform.
-
an iterative decision making scheme for markov decision processes and its application to self adaptive systems
Fundamental Approaches to Software Engineering, 2016Co-Authors: Taolue Chen, Yuan Feng, David S Rosenblum, P S ThiagarajanAbstract:Software is often governed by and thus adapts to phenomena that occur at runtime. Unlike traditional decision problems, where a decision-making model is determined for reasoning, the adaptation logic of such software is concerned with empirical data and is subject to practical constraints. We present an Iterative Decision-Making Scheme IDMS that infers both point and interval estimates for the undetermined transition probabilities in a Markov Decision Process MDP based on sampled data, and iteratively computes a confidently optimal scheduler from a given Finite Subset of schedulers. The most important feature of IDMS is the flexibility for adjusting the criterion of confident optimality and the sample size within the iteration, leading to a tradeoff between accuracy, data usage and computational overhead. We apply IDMS to an existing self-adaptation framework Rainbow and conduct a case study using a Rainbow system to demonstrate the flexibility of IDMS.
Nguyen Luc - One of the best experts on this subject based on the ideXlab platform.
-
Existence and uniqueness of Green's functions to nonlinear Yamabe problems
'Royal College of Obstetricians & Gynaecologists (RCOG)', 2021Co-Authors: Li Y, Nguyen LucAbstract:For a given Finite Subset S of a compact Riemannian manifold (M, g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M \ S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theorem
-
Existence and uniqueness of Green's functions to nonlinear Yamabe problems
2021Co-Authors: Li Yanyan, Nguyen LucAbstract:For a given Finite Subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theorem.Comment: Final version, to appear in Comm. Pure Appl. Mat
-
Existence and uniqueness of Green's functions to nonlinear Yamabe problems
2021Co-Authors: Li Yanyan, Nguyen LucAbstract:For a given Finite Subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theorem.Comment: To appear in Comm. Pure Appl. Mat
-
Existence and uniqueness of Green's function to a nonlinear Yamabe problem
2020Co-Authors: Li Yanyan, Nguyen LucAbstract:For a given Finite Subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of $S$ corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. We further show that these metrics arise as suitably rescaled limit for certain blow-up solutions to the corresponding nonlinear Yamabe problems. As a by-product, we define a purely local notion of Ricci lower bound for continuous metrics which are conformal to smooth metrics and prove a corresponding volume comparison theorem
