Cellular Decomposition Research Articles

Cellular decomposition and free resolution for split metacyclic spherical space forms

Given a free isometric action of a split metacyclic group on odd dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere equivariant with respect to the group action. A cell decomposition of the factor space and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group follow. In particular, the construction provides a simple explicit 4-periodic free resolution for the split metacyclic groups.

Multistable phase regulation for robust steady and transitional legged gaits

We develop robust methods that allow the specification, control, and transition of a multi-legged robot’s stepping pattern, its ‘gait’, during active locomotion over natural terrain. Resulting gaits emerge through the introduction of controllers that impose appropriately placed repellors within the space of gaits, the torus of relative leg phases, thereby mitigating against dangerous patterns of leg timing. Moreover, these repellors are organized with respect to a natural cellular decomposition of gait space and result in limit cycles with associated basins that are well characterized by these cells, thus conferring a symbolic character upon the overall behavioral repertoire. These ideas are particularly applicable to four- and six-legged robots, for which a large variety of interesting and useful (and, in many cases, familiar) gaits exist, and whose tradeoffs between speed and reliability motivate the desire for transitioning between them during active locomotion. We provide an empirical instance of this gait regulation scheme by application to a climbing hexapod, whose ‘physical layer’ sensor-feedback control requires adequate grasp of a climbing surface but whose closed-loop control perturbs the robot from its desired gait. We document how the regulation scheme secures the desired gait and permits operator selection of different gaits as required during active climbing on challenging surfaces.

Quiver Grassmannians and degenerate flag varieties

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group.

Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The Dirichlet problem

Consider a planar, bounded, m -connected region Ω, and let ∂ Ω be its boundary. Let 𝒯 be a cellular decomposition of Ω ∪ ∂ Ω, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair ( S , f ) where S is a genus ( m − 1) singular flat surface tiled by rectangles and f is an energy preserving mapping from 𝒯 (1) onto S . By a singular flat surface, we will mean a surface which carries a metric structure locally modeled on the Euclidean plane, except at a finite number of points. These points have cone singularities, and the cone angle is allowed to take any positive value (see for instance [28] for an excellent survey). Our realization may be considered as a discrete uniformization of planar bounded regions.

Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras

We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the "canonical basis" of cluster algebras of Kronecker type (found by Sherman and Zelevinsky) and of type $A_2^{(1)}$.

A fast algorithm to compute cohomology group generators of orientable 2-manifolds

In this paper a fast algorithm to compute cohomology group generators of cellular decomposition of any orientable open or closed 2-manifold is described. The presented algorithm is a dual version of algorithm to compute homology generators presented by David Eppstein in “Dynamic generators of topologically embedded graphs” and developed by Jeff Erickson and Kim Whittlesey in “Greedy optimal homotopy and homology generators”. Some parts of the paper bases on ideas presented in “Optimal discrete Morse functions for 2-manifolds” by Thomas Lewiner, Helio Lopes and Geovan Tavares. Extension of the algorithm to some non-manifold cases is provided.

Binomial Canonical Decompositions of Binomial Ideals

In this article, we prove that every binomial ideal in a polynomial ring over an algebraically closed field of characteristic zero admits a canonical primary decomposition into binomial ideals. Moreover, we prove that this special decomposition is obtained from a cellular decomposition which is also defined in a canonical way and does not depend on the field.

플로우 네트워크를 이용한 지능형 로봇의 경로계획

Many intelligent robots have to be given environmental information to perform tasks. In this paper an intelligent robot, that is, a cleaning robot used a sensor fusing method of two sensors: LRF and StarGazer, and then was able to obtain the information. Throughout wall following using laser displacement sensor, LRF, the working area is built during the robot turn one cycle around the area. After the process of wall following, a path planning which is able to execute the work effectively is established using flow network algorithm. This paper describes an algorithm for minimal turning complete coverage path planning for intelligent robots. This algorithm divides the whole working area by cellular decomposition, and then provides the path planning among the cells employing flow networks. It also provides specific path planning inside each cell guaranteeing the minimal turning of the robots. The proposed algorithm is applied to two different working areas, and verified that it is an optimal path planning method.

Lawson homology, morphic cohomology and Chow motives

AbstractIn this paper, Lawson homology and morphic cohomology are defined for Chow motives. As consequences, we rederive the projective bundle formula proved by Friedlander and Gabber, the blowup formula for Lawson homology by the first author, and a formula for certain homogeneous projective varieties, including those admitting cellular decompositions. We also define rational coefficient Lawson homology and morphic cohomology for Chow motives of finite quotient varieties. As a consequence, we obtain a formula for the Hilbert scheme of points on a smooth complex projective surface. We also discuss generic finite maps, in particular, we give examples of self‐product of smooth projective curves with nontrivial Griffiths groups by using a result of Ceresa. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Lamellar pyroxene-spinel symplectites in lunar olivine from the Luna 24 regolith

Cr-Ca lamellae in a magnesian olivine grain (section 1611) from the Luna 24 regolith were investigated in detail by electron microprobe analysis (EMPA), scanning electron microscopy (SEM), and transmission electron microscopy (TEM). It was found that the lamellae are parallel to the (100) plane of oxygen closest packing in olivine and consist of regular vermicular intergrowths of two phases, diopside (Di) and chromite (Chr), in the volume proportion Di: Chr ≈ 3: 1. The bulk chemical composition of the lamellae is approximated as Ca2Mg2Fe2+(Cr3+)2Si4O16. They are identical in phase composition to type A, F, and E symplectites from Apollo lunar samples [9]. Based on morphology and phase composition, the lamellar aggregates in the olivine grain from the Luna 24 regolith were classified as pyroxene (Px)-spinel (Spl) symplectites of a lamellar type, the formation of which was related to olivine oxidation at IW ≤ logfO2 ≤ QFM. The obtained data indicate a solid-phase mechanism of lamella formation and the existence of a lamellar precursor phase, which transformed subsequently into the Px-Spl symplectite. It was supposed that uvarovite-knorringite garnet produced by the oxidation of olivine at high pressures and t > 800°C could be the transitional phase during symplectite formation. The subsequent conversion of the garnet into the low-pressure assemblage of Px-Spl symplectites could occur via cellular decomposition in accordance with the reaction Ca2MgCr2Si3O12 + (Mg,Fe)2SiO4 = 2CaMgSi2O6 + FeCr2O4. The reported results are the first data of a detailed nanomineralogical investigation of lamellar Px-Spl symplectites in lunar olivine.

Boundary value problems on planar graphs and flat surfaces with integer cone singularities, II: The mixed Dirichlet–Neumann problem

In this paper we continue the study started in Hersonsky (in press) [16]. We consider a planar, bounded, m-connected region Ω, and let ∂ Ω be its boundary. Let T be a cellular decomposition of Ω ∪ ∂ Ω , where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair ( S , f ) where S is a special type of a (possibly immersed) genus ( m − 1 ) singular flat surface, tiled by rectangles and f is an energy preserving mapping from T ( 1 ) onto S. In Hersonsky (in press) [16] the solution of a Dirichlet problem defined on T ( 0 ) was utilized, in this paper we employ the solution of a mixed Dirichlet–Neumann problem.

Operator spin foam models

The goal of this paper is to introduce a systematic approach to spin foams. We define operator spin foams, that is foams labelled by group representations and operators, as our main tool. A set of moves we define in the set of the operator spin foams (among other operations) allows us to split the faces and the edges of the foams. We assign to each operator spin foam a contracted operator, by using the contractions at the vertices and suitably adjusted face amplitudes. The emergence of the face amplitudes is the consequence of assuming the invariance of the contracted operator with respect to the moves. Next, we define spin foam models and consider the class of models assumed to be symmetric with respect to the moves we have introduced, and assuming their partition functions (state sums) are defined by the contracted operators. Briefly speaking, those operator spin foam models are invariant with respect to the cellular decomposition, and are sensitive only to the topology and colouring of the foam. Imposing an extra symmetry leads to a family we call natural operator spin foam models. This symmetry, combined with assumed invariance with respect to the edge splitting move, determines a complete characterization of a general natural model. It can be obtained by applying arbitrary (quantum) constraints on an arbitrary BF spin foam model. In particular, imposing suitable constraints on a spin(4) BF spin foam model is exactly the way we tend to view 4D quantum gravity, starting with the BC model and continuing with the Engle–Pereira–Rovelli–Livine (EPRL) or Freidel–Krasnov (FK) models. That makes our framework directly applicable to those models. Specifically, our operator spin foam framework can be translated into the language of spin foams and partition functions. Among our natural spin foam models there are the BF spin foam model, the BC model, and a model corresponding to the EPRL intertwiners. Our operator spin foam framework can also be used for more general spin foam models which are not symmetric with respect to one or more moves we consider.

The H-polynomial of an irreducible representation

Let G be a simple algebraic group. Associated with the finite-dimensional rational representation ρ : G → End ( V ) of G there is the monoid M ρ = K ⁎ ρ ( G ) ¯ ⊆ End ( V ) and the projective G × G -embedding P ρ = [ M ρ ∖ { 0 } ] / K ⁎ . One can identify the cases where P ρ is rationally smooth; and in such cases it is desirable to calculate the H-polynomial, H, of P ρ . In this paper we consider the situation where ρ is irreducible. We then determine H explicitly in terms of combinatorial invariants of ρ. Indeed, there is a canonical cellular decomposition for P ρ . These cells are defined in terms of idempotents, B × B -orbits and other natural quantities obtained from M ρ . Furthermore, H is obtained by recording the dimension of each of these cells in terms of the descent system of M ρ . As a special case we reacquire the well-known formula for the Poincaré polynomial of a “wonderful embedding” of a simple algebraic group of adjoint type.

Coherent states for FLRW space-times in loop quantum gravity

We construct a class of coherent spin-network states that capture properties of curved space-times of the Friedmann-Lama\^{\i}tre-Robertson-Walker type on which they are peaked. The data coded by a coherent state are associated to a cellular decomposition of a spatial ($t=\mathrm{const}$) section with a dual graph given by the complete five-vertex graph, though the construction can be easily generalized to other graphs. The labels of coherent states are complex $SL(2,\mathbb{C})$ variables, one for each link of the graph, and are computed through a smearing process starting from a continuum extrinsic and intrinsic geometry of the canonical surface. The construction covers both Euclidean and Lorentzian signatures; in the Euclidean case and in the limit of flat space we reproduce the simplicial 4-simplex semiclassical states used in spin foams.

Self‐dual and self‐petrie‐dual regular maps

AbstractRegular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a2 = b2 = c2 = (ab)k = (bc)m = (ca)2 = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b↦b, c↦a and a↦c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012

Exploiting hierarchical parallelisms for molecular dynamics simulation on multicore clusters

We have developed a scalable hierarchical parallelization scheme for molecular dynamics (MD) simulation on multicore clusters. The scheme explores multilevel parallelism combining: (1) Internode parallelism using spatial decomposition via message passing; (2) intercore parallelism using cellular decomposition via multithreading employing a master/worker model; (3) data-level optimization via single-instruction multiple-data (SIMD) parallelism with various code transformation techniques. By using a hierarchy of parallelisms, the scheme exposes very high concurrency and data locality, thereby achieving: (1) internode weak-scaling parallel efficiency 0.985 on 106,496 BlueGene/L nodes (0.975 on 32,768 BlueGene/P nodes), internode strong-scaling parallel efficiency 0.90 on 8,192 BlueGene/L nodes; (2) intercore multithread parallel efficiency 0.65 for eight threads on a dual quadcore Xeon platform; and (3) SIMD speedup around 2 for problem sizes ranging from 3,072 to 98,304 atoms. Furthermore, the effect of memory-access penalty on SIMD performance is analyzed, and an application-based SIMD analysis scheme is proposed to help programmers determine whether their applications are amenable to SIMDization.

Coverage path planning for UAVs based on enhanced exact cellular decomposition method

In this paper, an enhanced exact cellular decomposition method to plan the coverage path of UAVs in a polygon area is proposed. To be more specific, the contributions of the paper are: firstly, the turning motion of UAVs is shown to be less efficient from the viewpoints of route length, duration and energy. Secondly, the problem of coverage Path Planning (CPP) in a convex polygon area is transformed to width calculation of the convex polygon, and a novel algorithm to calculate the widths of convex polygons with time complexity of O( n) is developed. The path of the least number of turns for an UAV based on the widths of convex polygons is devised. Thirdly, a convex decomposition algorithm for minimum width sum based on the greedy recursive method which revolves around decomposing the concave area into convex subregions is developed. It is proved that the algorithm is a polynomial time algorithm. To avoid unnecessary back and forth motion, some entirely adjacent subregions are combined. Finally, comparing different weights of two joint-points, a subregion connection algorithm based on minimum traversal of weighted undirected graph is proposed to connect the coverage paths of the subregions. Simulation results show that the proposed method is feasible and effective.

A Role for Serglycin Proteoglycan in Mast Cell Apoptosis Induced by a Secretory Granule-mediated Pathway

Mast cell secretory granules (secretory lysosomes) contain large amounts of fully active proteases bound to serglycin proteoglycan. Damage to the granule membrane will thus lead to the release of serglycin and serglycin-bound proteases into the cytosol, which potentially could lead to proteolytic activation of cytosolic pro-apoptotic compounds. We therefore hypothesized that mast cells are susceptible to apoptosis induced by permeabilization of the granule membrane and that this process is serglycin-dependent. Indeed, we show that wild-type mast cells are highly sensitive to apoptosis induced by granule permeabilization, whereas serglycin-deficient cells are largely resistant. The reduced sensitivity of serglycin(-/-) cells to apoptosis was accompanied by reduced granule damage, reduced release of proteases into the cytosol, and defective caspase-3 activation. Mechanistically, the apoptosis-promoting effect of serglycin involved serglycin-dependent proteases, as indicated by reduced sensitivity to apoptosis and reduced caspase-3 activation in cells lacking individual mast cell-specific proteases. Together, these findings implicate serglycin proteoglycan as a novel player in mast cell apoptosis.

Integrated On-Line Localization, Mapping and Coverage Algorithm of Unknown Environments for Robotic Vacuum Cleaners Based on Minimal Sensing

This paper presents a new complete coverage algorithm of a robotic vacuum cleaner (RVC) with a low-cost sensor in an unknown environment. To achieve complete coverage, the RVC must have navigation systems for precise position estimation with localization and a prior map or a map using information that has been continuously collected from the environment. To do this, two-dimensional laser range finders and vision sensors are becoming increasingly popular in mobile robotics, and various methods using sensors like these have been introduced by many researchers. However, it is difficult to apply the methods to sensors used in most RVCs due to their constraints. In this paper, we present a new method applied to most RVCs. For developing the method, we considered the two main problems of maintaining low computational load, and coping with low-cost sensor systems with limited range, detection uncertainty and measurement error. To solve the problems, we propose an assumption that major structures of an indoor environment are rectilinear, and can be represented by sets of parallel and perpendicular lines. Then we derive an algorithm that uses this assumption to map the environment, localize the robot and plan the coverage path with a new cellular decomposition approach. Simulation and experiments verify that the proposed method guarantees complete coverage.