PL Manifolds Research Articles

Metric Ricci Curvature and Flow for PL Manifolds

We summarize here the main ideas and results of our papers [28], [14], as presented at the 2013 CIRM Meeting on Discrete curvature and we augment these by bringing up an application of one of our main results, namely to solving a problem regarding cube complexes.

Faithful tropicalization of the Grassmannian of planes

We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich’s sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. As a consequence, we prove that the tropical multiplicities of all points in the tropical projective Grassmannian are equal to one. Finally, we determine a piecewise linear structure on the image of our section that corresponds to the polyhedral structure on the tropical projective Grassmannian.

Triangulation of the map of a G-manifold to its orbit space

AbstractLet G be a Lie group, and let M be a smooth proper G-manifold. Let M/G denote the orbit space, and let π : M → M/G be the natural map. It is known that M/G is homeomorphic to a polyhedron. In the present paper we show that there exist a piecewise linear (PL) manifold P, a polyhedron L, and homeomorphisms τ : P → M and σ : M/G → L such that σ o π o τ is PL. This is an application of the theory of subanalytic sets and subanalytic maps of Shiota. If M and the G-action are, moreover, subanalytic, then we can choose τ and σ subanalytic and P and L unique up to PL homeomorphisms.

Canonical embeddings of [formula omitted] into orientable n-dimensional closed PL manifolds for [formula omitted

For each embedded oriented circle c in an n-dimensional closed PL manifold M, we define a canonical embedding with respect to a triangulation of M, of S1×Δn−1 into a regular neighborhood of the embedded circle in M. For n>4, we give a necessary and sufficient condition for an embedding of a surface with boundary in M, such that the embedding together with the canonical embeddings on the boundary components of the surface, extends to an embedding of the regular neighborhood of the surface in Rn to a regular neighborhood of the embedded surface in M.

Markov Decision Processes with Threshold Based Piecewise Linear Optimal Policies

This letter investigates the structure of the optimal policy for a class of Markov decision processes (MDPs), having convex and piecewise linear cost function. The optimal policy is proved to have a piecewise linear structure that alternates flat and constant-slope pieces, resembling a staircase with tilted rises and as many steps (thresholds) as the breakpoints of the cost function. This particular structure makes it possible to express the policy in a very compact manner, particularly suitable to be stored in low-end devices. More importantly, the threshold-based form of the optimal policy can be exploited to reduce the computational complexity of the iterative dynamic programming (DP) algorithm used to solve the problem. These results apply to a rather wide set of optimization problems, typically involving the management of a resource buffer such as the energy stored in a battery, or the packets queued in a wireless node.

S-limit shadowing is [formula omitted]-dense

We prove (see Theorem 1) that the set of maps (resp. surjective maps) with the shadowing property is C0-residual in the space of all continuous maps (resp. continuous surjective maps) of a compact topological manifold (possibly with boundary), which extends the similar result known for homeomorphisms (Pilyugin and Plamenevskaya, 1999) [22], as well as generalize the analogous result for maps (Kos̀cielniak et al., Preprint 2013) [12], obtained under an additional assumption that the manifold admits some kind of a piecewise linear structure. Moreover, we prove (see Theorem 2) that the set of maps (resp. surjective maps) with the s-limit shadowing property is C0-dense in the space of all continuous maps (resp. continuous surjective maps).

Coloured graphs representing PL 4-manifolds

Abstract Crystallization theory is a representation method for compact PL manifolds by means of a particular class of edge-coloured graphs. The combinatorial nature of this representation allows to elaborate and implement algorithmic procedures for the generation and analysis of catalogues of closed PL n-manifolds. In this paper we discuss the concepts which are involved in these procedures for n = 4 and present classification results arising from the study of the initial segment of the catalogue.

ESPACES DE BERKOVICH, POLYTOPES, SQUELETTES ET THÉORIE DES MODÈLES

Let X be an analytic space over a non-Archimedean, complete field k and let f = (f1,…,fn) be a family of invertible functions on X. Let us recall two results. (1) If X is compact, the compact set |f|(X) is a polytope of the ℝ-vector space (we use the multiplicative notation); this is due to Berkovich in the locally algebraic case (his proof made use of de Jong's alterations), and has been extended to the general case by the author. The locally algebraic case could also have been deduced quite formally from a former result by Bieri and Groves, based upon explicit computations on Newton polygons. (2) If moreover X is Hausdorff and n-dimensional, and if φ denotes the morphism induced by f, then the pre-image of the skeleton Sn of under φ has a piecewise-linear structure making φ-1(Sn) → Sn a piecewise immersion; this is due to the author, and his proof also made use of de Jong's alterations. In this article, we improve (1) and (2), and give new proofs of both of them. Our proofs are based upon the model theory of algebraically closed, nontrivially valued fields and do not involve de Jong's alterations. Let us quickly explain what we mean by improving (1) and (2). • Concerning (1), we also prove that if x ∈ X, there exists a compact analytic neighborhood U of x, such that for every compact analytic neighborhood V of x in X, the germs of polytopes (|f|(V), |f|(x)) and (|f|(U), |f|(x)) coincide. • Concerning (2), we prove that the piecewise linear structure on φ-1(Sn) is canonical, that is, does not depend on the map we choose to write it as a pre-image of the skeleton; we thus answer a question which was asked to us by Temkin. Moreover, we prove that the pre-image of the skeleton stabilizes after a finite, separable ground field extension, and that if φ1,…,φm are finitely many morphisms from X to , the union also inherits a canonical piecewise-linear structure.

Hinging hyperplane based regression tree identified by fuzzy clustering and its application

Hierarchical fuzzy modeling techniques have great advantage since model accuracy and complexity can be easily controlled thanks to the transparent model structures. A novel tool for regression tree identification is proposed based on the synergistic combination of fuzzy c-regression clustering and the concept of hierarchical modeling. In a special case (c=2), fuzzy c-regression clustering can be used for identification of hinging hyperplane models. The proposed method recursively identifies a hinging hyperplane model that contains two linear submodels by partitioning operating region of one local linear model resulting a binary regression tree. Novel measures of model performance and complexity are developed to support the analysis and building of the proposed special model structure. Effectiveness of proposed model is demonstrated by benchmark regression datasets. Examples also demonstrate that the proposed model can effectively represent nonlinear dynamical systems. Thanks to the piecewise linear model structure the resulted regression tree can be easily utilized in model predictive control. A detailed application example related to the model predictive control of a water heater demonstrate that the proposed framework can be effectively used in modeling and control of dynamical systems.

On piecewise linear cell decompositions

In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain "elementary" moves. This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.

Highly Complex Chaotic System with Piecewise Linear Nonlinearity and Compound Structures

A new chaotic system is presented using a single parameter for a two-scroll attractor with high complexity, high chaoticity and widely chaotic range. The system employs two quadratic nonlinearities and two piecewise-linear nonlinearities. The high chaoticity is measured by the the maximum Lyapunov Exponent of 0.429 and the high complexity is measured by the Kaplan-Yorke dimension of 2.3004. Dynamic properties are described in terms of symmetry, a dissipative system, an existence of attractor, equilibria, Jacobian matrices, bifurcations, Poincaroe maps, chaotic waveforms, chaotic spectrum, and forming mechanisms of compound structures .

Relations in Grassmann Algebra Corresponding to Three- and Four-Dimensional Pachner Moves

New algebraic relations are presented, involving anticommuting Grassmann variables and Berezin integral, and corresponding naturally to Pachner moves in three and four dimensions. These relations have been found experimentally - using symbolic computer calculations; their essential new feature is that, although they can be treated as deformations of relations corresponding to torsions of acyclic complexes, they can no longer be explained in such terms. In the simpler case of three dimensions, we define an invariant, based on our relations, of a piecewise-linear manifold with triangulated boundary, and present example calculations confirming its nontriviality.

A production-transportation problem with piecewise linear cost structures

A production-transportation problem with piecewise linear cost structures

REGULAR GENUS AND PRODUCTS OF SPHERES

A crystallization of a closed connected PL manifold M is a special edge-colored graph representing M via a contracted triangulation. The regular genus of M is the minimum genus of a closed connected surface into which a crystallization of M regularly embeds. We disprove a conjecture on the regular genus of <TEX>$\mathbb{S}\;{\times}\;\mathbb{S}^n$</TEX>, <TEX>$n\;{\geq}\;3$</TEX>, stated in [J. Korean Math. Soc. 41 (2004), no. 3, p. 420].

From the Icosahedron to Natural Triangulations of ℂP 2 and S 2×S 2

We present two constructions in this paper: (a) a 10-vertex triangulation $\mathbb{C}P^{2}_{10}$ of the complex projective plane ℂP 2 as a subcomplex of the join of the standard sphere ($S^{2}_{4}$ ) and the standard real projective plane ($\mathbb{R}P^{2}_{6}$, the decahedron), its automorphism group is A 4; (b) a 12-vertex triangulation (S 2×S 2)12 of S 2×S 2 with automorphism group 2S 5, the Schur double cover of the symmetric group S 5. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S 2×S 2. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that ℂP 2 has S 2×S 2 as a two-fold branched cover; we construct the triangulation $\mathbb{C}P^{2}_{10}$ of ℂP 2 by presenting a simplicial realization of this covering map S 2×S 2→ℂP 2. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S 2×S 2, different from the triangulation alluded to in (b). This gives a new proof that Kuhnel’s $\mathbb{C}P^{2}_{9}$ triangulates ℂP 2. It is also shown that $\mathbb{C}P^{2}_{10}$ and (S 2×S 2)12 induce the standard piecewise linear structure on ℂP 2 and S 2×S 2 respectively.

Complexity of PL manifolds

We extend Matveev’s complexity of 3‐manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant). Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with nonpositive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4‐manifolds of complexity zero (manifolds without 3‐handles, doubles of 2‐handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group). 57Q99; 57M99

MANIFOLDS WITH TRIVIAL HOMOLOGY GROUPS IN SOME RANGE AS CODIMENSION-K FIBRATORS

Approximate fibrations provide a useful class of maps. Fibrators give instant detection of maps in this class, and PL fibrators do the same in the PL category. We show that rational homology spheres with some additional conditions are codimension-k PL fibrators and PL manifolds with trivial homology groups in some range can be codimension-k (k > 2) PL fibrators.

An Algorithm for Finding Candidate Synaptic Sites in Computer Generated Networks of Neurons with Realistic Morphologies

Neurons make synaptic connections at locations where axons and dendrites are sufficiently close in space. Typically the required proximity is based on the dimensions of dendritic spines and axonal boutons. Based on this principle one can search those locations in networks formed by reconstructed neurons or computer generated neurons. Candidate synapses are then located where axons and dendrites are within a given criterion distance from each other. Both experimentally reconstructed and model generated neurons are usually represented morphologically by piecewise-linear structures (line pieces or cylinders). Proximity tests are then performed on all pairs of line pieces from both axonal and dendritic branches. Applying just a test on the distance between line pieces may result in local clusters of synaptic sites when more than one pair of nearby line pieces from axonal and dendritic branches is sufficient close, and may introduce a dependency on the length scale of the individual line pieces. The present paper describes a new algorithm for defining locations of candidate synapses which is based on the crossing requirement of a line piece pair, while the length of the orthogonal distance between the line pieces is subjected to the distance criterion for testing 3D proximity.

Synchronization methods and properties of the improved chaotic system with constant Lyapunov exponent spectrum

The existence of special piecewise-linear structure, global linear amplitude adjuster and phase-reverse controller in improved chaotic system with constant Lyapunov exponent spectrum contributed much to finding new synchronization methods and new adjusting parameters. According to the principle of generalized synchronization, by constructing appropriate driving system and response system, generalized synchronization of the improved chaotic system with constant Lyapunov exponent spectrum then can be achieved. By changing the parameter of response system, the complete synchronization and generalized projective synchronization can be further realized. The amplitude of the state variable of the driving system and the response system can be controlled to increase and decrease synchronously by the global linear amplitude adjuster, while the phase of certain variables can be controlled to reverse synchronously by the phase-reverser. This synchronization system dispenses with other special controllers, possesses simple structure and so can be implemented easily. Finally, the implement circuit of synchronization system is designed. It is validated by experiment simulation that the synchronization methods are feasible, the amplitude and the phase of the state variable in the synchronization system can be controlled by special parameters of the improved chaotic system.

The van Kampen obstruction and its relatives

We review a cochain-free treatment of the classical van Kampen obstruction \theta to embeddability of an n-polyhedron into R^{2n} and consider several analogues and generalizations of \theta, including an extraordinary lift of \theta which in the manifold case has been studied by J.-P. Dax. The following results are obtained. - The mod 2 reduction of \theta is incomplete, which answers a question of Sarkaria. - An odd-dimensional analogue of \theta is a complete obstruction to linkless embeddability (="intrinsic unlinking") of the given n-polyhedron in R^{2n+1}. - A "blown up" 1-parameter version of \theta is a universal type 1 invariant of singular knots, i.e. knots in R^3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (=Polyak-Viro) formula. - Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, non-embeddable in a given R^m, m\ge 3(n+1)/2, contains a subset X such that no map N\to R^m sends X and N-X to disjoint sets. - We elaborate on McCrory's analysis of the Zeeman spectral sequence to geometrically characterize "k-co-connected and locally k-co-connected" polyhedra, which we embed in R^{2n-k} for k<(n-3)/2 extending the Penrose-Whitehead-Zeeman theorem.