PL Manifolds Research Articles

Penalized Triograms: Total Variation Regularization for Bivariate Smoothing

SummaryHansen, Kooperberg and Sardy introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modelling of bivariate densities and regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are more commonly used in statistical applications. Triograms employ basis functions consisting of linear ‘tent functions’ defined with respect to a triangulation of a given planar domain. As in knot selection for univariate splines, Hansen and colleagues adopted the regression spline approach of Stone. Vertices of the triangulation are introduced or removed sequentially in an effort to balance fidelity to the data and parsimony. We explore a smoothing spline variant of the triogram model based on a roughness penalty adapted to the piecewise linear structure of the triogram model. We show that the roughness penalty proposed may be interpreted as a total variation penalty on the gradient of the fitted function. The methods are illustrated with real and artificial examples, including an application to estimated quantile surfaces of land value in the Chicago metropolitan area.

A greedy Delaunay-based surface reconstruction algorithm

In this paper, we present a new greedy algorithm for surface reconstruction from unorganized point sets. Starting from a seed facet, a piecewise linear surface is grown by adding Delaunay triangles one by one. The most plausible triangles are added first and in such a way as to prevent the appearance of topological singularities. The output is thus guaranteed to be a piecewise linear orientable manifold, possibly with boundary. Experiments show that this method is very fast and achieves topologically correct reconstruction in most cases. Moreover, it can handle surfaces with complex topology, boundaries, and nonuniform sampling.

The homotopy type of the matroid grassmannian

Stiefel [26] and Whitney [28] and with integral coefficients by Pontrjagin [24], make up the primary source of first-order invariants of smooth manifolds. When their utility was first recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques. Modulo 2, this hope was realized rather quickly: it is not hard to see that the Stiefel-Whitney classes are PL invariants. Moreover, Whitney was able to produce a simple explicit formula for the class in codimension i in terms of the i-skeleton of the barycentric subdivision of a triangulated manifold (for a proof of this result, see [13]). One would like to find an analogue of these results for the Pontrjagin classes. However, such a naive goal is entirely out of reach; indeed, Milnor's use of the Pontrjagin classes to construct an invariant which distinguishes between nondiffeomorphic manifolds which are homeomorphic and PL isomorphic

Models and Methods for Merge-in-Transit Operations

We develop integer programming formulations and solution methods for addressing operational issues in merge-in-transit distribution systems. The models account for various complex problem features, including the integration of inventory and transportation decisions, the dynamic and multimodal components of the application, and the nonconvex piecewise linear structure of the cost functions. To accurately model the cost functions, we introduce disaggregation techniques that allow us to derive a hierarchy of linear programming relaxations. To solve these relaxations, we propose a cutting-plane procedure that combines constraint and variable generation with rounding and branch-and-bound heuristics. We demonstrate the effectiveness of this approach on a large set of test problems with instances derived from actual data from the computer industry that contain almost 500,000 integer variables.

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III. Moves 1 ↔ 5 and Related Structures

We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. First, we write out formulas for moves 3 -> 3 and 2 <-> 4 based on the results of our two previous works, and then we study in detail moves 1 <-> 5. On this basis, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for sphere S^4; in particular, the complex turns out, indeed, to be acyclic.

On the Effectiveness of Zero-Inventory-Ordering Policies for the Economic Lot-Sizing Model with a Class of Piecewise Linear Cost Structures

We consider an economic lot-sizing problem with a special class of piecewise linear ordering costs, which we refer to as the class of modified all-unit discount cost functions. Such an ordering cost function represents transportation costs charged by many less-than truckload carriers. We show that even special cases of the lot-sizing problem are NP-hard and therefore analyze the effectiveness of easily implementable policies. In particular, we demonstrate that there exists a zero-inventory-ordering(ZIO) policy, i.e., a policy in which an order is placed only when the inventory level drops to zero, whose total inventory and ordering cost is no more than 4/3 times the optimal cost. Furthermore, if the ordering cost function does not vary over time, then the cost of the best ZIO policy is no more than 5.6/4.6 times the optimal cost. These results hold for any transportation and holding cost functions that satisfy the following properties: (i) they are non decreasing functions, and (ii) the associated cost per unit is non increasing. Finally, we report on a numerical study that shows the effectiveness of ZIO policies on a set of test problems.

Effective Zero-Inventory-Ordering Policies for the Single-Warehouse Multiretailer Problem with Piecewise Linear Cost Structures

We analyze the problem faced by companies that rely on TL (Truckload) and LTL (Less than Truckload) carriers for the distribution of products across their supply chain. Our goal is to design simple inventory policies and transportation strategies to satisfy time varying demands over a finite horizon, while minimizing system wide cost by taking advantage of quantity discounts in the transportation cost structures. For this purpose, we study the cost effectiveness of restricting the inventory policies to the class of zero-inventory-ordering (ZIO) policies in a single-warehouse multiretailer scenario in which the warehouse serves as a cross-dock facility. In particular, we demonstrate that there exists a ZIO inventory policy whose total inventory and transportation cost is no more than 4/3 (5.6/4.6 if transportation costs are stationary) times the optimal cost. However, finding the best ZIO policy is an NP hard problem as well. Thus, we propose two algorithms to find an effective ZIO policy: An exact algorithm whose running time is polynomial for any fixed number of retailers, and a linear-programming-based heuristic whose effectiveness is demonstrated in a series of computational experiments. Finally, we extend the worst-case results developed in this paper to systems in which the warehouse does hold inventory.

On Poincaré four-complexes with free fundamental groups

Let X be a closed connected and oriented PL manifold whose fundamental group p1 is a free group of rank p. Let L be the integral group ring of p1. Then H2ðX ; LÞ is L-free (see (6)). We show that there is a bijective correspondence between homotopy equivalence classes of 4-dimensional Poincarecomplexes Y with Y ð3Þ G X ð3Þ and invertible hermitian matrices of type k over L, where k is the rank of H2ðX ; LÞ.

Disjoint spine phenomena in certain contractible n-manifolds (n⩾5)

If M is a compact PL manifold with boundary containing a subpolyhedron K in its interior, then K is said to be a PL spine of M provided M collapses to K (M↘K). Guilbault [Topology 34 (1) (1995) 99–108] has shown that certain nontrivial contractible manifolds possess disjoint spines. His results stem from a standing conjecture regarding disjoint spines in contractible 4-manifolds constructed by Mazur. More to the point, there is a dimensional requirement introduced by his techniques; Guilbault produces such manifolds in dimensions n⩾9. We shall provide techniques which allow the construction of examples in dimensions n⩾5 following the path laid out by Guilbault. The new techniques will provide a slight strengthening of some other Guilbault results as well.

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: I. Moves 3→3

We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on the usage of Euclidean geometric values.

Invariants of PL Manifolds from Metrized Simplicial Complexes. Three-Dimensional Case

An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.

L'invariant η Pour Les Variétés Lipschitziennes

The η-invariant has been defined for C∞-manifolds by M.F. Atiyah, V.K. Patodi and I.M. Singer, and more recently for manifolds with corners by A. Hassell, R. Mazzeo and R.B. Melrose, and for stratified PL manifolds by H. Moscovici and F.B. Wu. In the present work, this invariant is generalized in the framework of lipschitz riemannian manifolds. This involves selfadjoint extensions of the signature operator on a lipschitz manifold with boundary, and measurable differential forms which represent the Pontryagyn classes of the manifold. This allows us to extend from smooth to topological manifolds the Atiyah-Patodi-Singer index theorem for flat bundles.

Nonlinear adaptive control using networks of piecewise linear approximators

This paper presents a stable nonparametric adaptive control approach using a piecewise local linear approximator. The continuous piecewise linear approximator is developed and its universal approximation capability is proved. The controller architecture is based on adaptive feedback linearization plus sliding mode control. A time varying activation region is introduced for efficient self-organization of the approximator during operation.We modify the adaptive control approach for piecewise linear approximation and self-organizing structures. In addition, we provide analyses of asymptotic stability of the tracking error and parameter convergence for the proposed adaptive control scheme with the on-line self-organizing structure. The method with a deadzone is also discussed to prevent a high-frequency input which might excite the unmodeled dynamics in practical applications. The application of the piecewise linear adaptive control method is demonstrated by a computational simulation.

A. D. Alexandrov's length manifolds with one-sided bounded curvature

The paper is devoted to discovering relations between inner metrics of manifolds with sectional curvature bounded from above (from below) by some constant and topological (simplicial, piecewise-linear, or smooth) structures on such manifolds. Examples of Alexandrov spaces with one-sided bounded curvature having exotic properties are given. Bibliography: 6 titles.

TOPOLOGY OF COMBINATORIAL DIFFERENTIAL MANIFOLDS

We prove that all combinatorial differential manifolds involving only Euclidean oriented matroids are PL manifolds. In doing so we introduce a new notion of triangulations of oriented matroids, cand show that any triangulation of a Euclidean oriented matroid is a PL sphere. In Section 5 we adapt these results to get a new definition of triangulations of oriented matroid polytopes, and show that any triangulation of a Euclidean oriented matroid polytope is a PL ball.

A new mixed integer formulation for the maximum regret problem

The minimax regret solution to a linear program with interval objective function coefficients can be found using an algorithm that, at each iteration, solves a linear program to generate a candidate solution and a mixed integer program to find the corresponding maximum regret. This paper presents a new formulation for the latter problem that exploits the piecewise linear structure of the cost coefficients. Computational results indicate that this yields a stronger formulation than the previous approach.

A New Mixed Integer Formulation for the Maximum Regret Problem

The minimax regret solution to a linear program with interval objective function coefficients can be found using an algorithm that, at each iteration, solves a linear program to generate a candidate solution and a mixed integer program to find the corresponding maximum regret. This paper presents a new formulation for the latter problem that exploits the piecewise linear structure of the cost coefficients. Computational results indicate that this yields a stronger formulation than the previous approach.

Open problems on graphs arising from geometric topology

We have collected several open problems on graphs which arise in geometric topology, in particular in the following areas: 1. (1) basic embeddability of compacta into the plane R 2; 2. (2) approximability of maps by embeddings; 3. (3) uncountable collections of continua in R 2 and their span; and 4. (4) representations of closed PL manifolds by colored graphs. These problems should be of interest to both topologists and combinatorists.

Variations of the boundary geometry of $3$-dimensional hyperbolic convex cores

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its bending is described by a measured geodesic lamination. We show that, as the hyperbolic metric of the 3--manifold M varies differentiably, the hyperbolic metric of the surface S varies in a C^1, but usually not C^2, manner. Differentiability properties for the bending measured lamination are conceptually less simple, because the space ML(S) of measured geodesic laminations on S has no natural differentiable structure. However, ML(S) is a piecewise linear manifold, and we also show that the bending measured geodesic lamination varies differentiably in a piecewise linear sense. The two results are proved simultaneously, mixing the differentiable and piecewise linear contexts. In particular, the 'corners' of the piecewise linear structure of ML(S) account for the fact that the metric of S is not a C^2 function of the metric of M. A by-product of the proof is a similar differentiablity property for Thurston's parametrization of complex projective structures on a surface by hyperbolic metrics and measured laminations.

A combinatorial approach to diffeomorphism invariant quantum gauge theories

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.