A note on K4-closures in Hamiltonian graph theory

Let G=(V, E) be a 2-connected graph. We call two vertices u and v of G a K4-pair if u and v are the vertices of degree two of an induced subgraph of G which is isomorphic to K4 minus an edge. Let x and y be the common neighbors of a K4-pair u, v in an induced K4−e. We prove the following result: If N(x)N(y)N(u)N(v){u,v}, then G is hamiltonian if and only if G+uv is h amiltonian. As a consequence, a claw-free graph G is hamiltonian if and only if G+uv is hamiltonian, where u,v is a K4-pair. Based on these results we define socalled K4-closures of G. We give infinite classes of graphs with small maximum degree and large diameter, and with many vertices of degree two having complete K4-closures

Graph-based SLAM-Aware Exploration with Prior Topo-Metric Information

Autonomous exploration requires the robot to explore an unknown environment while constructing an accurate map with the SLAM (Simultaneous Localization and Mapping) techniques. Without prior information, the exploratory performance is usually conservative due to the limited planning horizon. This paper exploits a prior topo-metric graph of the environment to benefit both the exploration efficiency and the pose graph accuracy in SLAM. Based on recent advancements in relating pose graph reliability with graph topology, we are able to formulate both objectives into a SLAM-aware path planning problem over the prior graph, which finds a fast exploration path with informative loop closures that globally stabilize the pose graph. Furthermore, we derive theoretical thresholds to speed up the greedy algorithm to the problem, which significantly prune non-optimal loop closures in iterations. The proposed planner is incorporated into a hierarchical exploration framework, with flexible features including path replanning and online prior map update that adds additional information to the prior graph. Extensive experiments indicate that our method has comparable exploration efficiency to others while consistently maintaining higher mapping accuracy in various environments. Our implementations will be open-source on GitHub.Comment: 8 pages, 6 figure

Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

The Jones polynomial: quantum algorithms and applications in quantum complexity theory

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.Comment: 34 pages. Substantial revision. Increased emphasis on HOMFLYPT, greatly simplified arguments and improved organizatio

Equivariant Intersection Cohomology of BXB Orbit Closures in the Wonderful Compactification of a Group

This thesis studies the topology of a particularly nice compactification that exists for semisimple adjoint algebraic groups: the wonderful compactification. The compactifica- tion is equivariant, extending the left and right action of the group on itself, and we focus on the local and global topology of the closures of Borel orbits. It is natural to study the topology of these orbit closures since the study of the topology of Borel orbit closures in the flag variety (that is, Schubert varieties) has proved to be inter- esting, linking geometry and representation theory since the local intersection cohomology Betti numbers turned out to be the coefficients of Kazhdan-Lusztig polynomials. We compute equivariant intersection cohomology with respect to a torus action because such actions often have convenient localization properties enabling us to use data from the moment graph (roughly speaking the collection of 0 and 1-dimensional orbits) to compute the equivariant (intersection) cohomology of the whole space, an approach commonly re- ferred to as GKM theory after Goresky, Kottowitz and MacPherson. Furthermore in the GKM setting we can recover ordinary intersection cohomology from the equivariant inter- section cohomology. Unfortunately the GKM theorems are not practical when computing intersection cohomology since for singular varieties we may not a priori know the local equivariant intersection cohomology at the torus fixed points. Braden and MacPherson address this problem, showing how to algorithmically apply GKM theory to compute the equivariant intersection cohomology for a large class of varieties that includes Schubert varieties. Our setting is more complicated than that of Braden and MacPherson in that we must use some larger torus orbits than just the 0 and 1-dimensional orbits. Nonetheless we are able to extend the moment graph approach of Braden and MacPherson. We define a more general notion of moment graph and identify canonical sheaves on the generalized moment graph whose sections are the equivariant intersection cohomology of the Borel orbit closures of the wonderful compactification