PL Manifolds Research Articles

Free Fermions on a Piecewise Linear Four-Manifold. II: Pachner Moves

This is the second in a series of papers where we construct an invariant of a four-dimensional piecewise linear manifold $M$ with a given middle cohomology class $h\in H^2(M,\mathbb C)$. This invariant is the square root of the torsion of unusual chain complex introduced in Part I (arXiv:1605.06498) of our work, multiplied by a correcting factor. Here we find this factor by studying the behavior of our construction under all four-dimensional Pachner moves, and show that it can be represented in a multiplicative form: a product of same-type multipliers over all 2-faces, multiplied by a product of same-type multipliers over all pentachora.

Study of a smart platform based on backstepping control method

A structural model is significant for the verification of structural control algorithms. However, for nonlinear behavior, experiments are mostly destructive tests that are costly, and conducting repetitive structural experiments is difficult. Therefore, a repetitive structural vibration model is important for structural vibration control. In this study, a smart platform to realize different structural behaviors is developed based on the backstepping control algorithm. Lyapunov functions are used to derive the control law. Simulations show that the designed model can track the structural responses of different arbitrary linear structures very well. In addition, the proposed platform can track responses of different piecewise linear structures and desired models with various hysteresis very well. Numerical results verify the effectiveness of the proposed tracking controller through the backstepping method for the established platform.

A general state-sum construction of 2-dimensional topological quantum field theories with defects

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142 ]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

Equivariant smoothing of piecewise linear manifolds

AbstractWe prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly.

Genus-minimal crystallizations of PL 4-manifolds

For $d\geq 2$, the regular genus of a closed connected PL $d$-manifold $M$ is the least genus (resp., half of the genus) of an orientable (resp., a non-orientable) surface into which a crystallization of $M$ imbeds regularly. The regular genus of every orientable surface equals its genus, and the regular genus of every 3-manifold equals its Heegaard genus. For every closed connected PL $4$-manifold $M$, it is known that its regular genus $\mathcal G(M)$ is at least $2 \chi (M) + 5m -4$, where $m$ is the rank of the fundamental group of $M$. In this article, we introduce the concept of "weak semi-simple crystallization" for every closed connected PL $4$-manifold $M$, and prove that $\mathcal G(M)= 2 \chi (M) + 5m -4$ if and only if $M$ admits a weak semi-simple crystallization. We then show that the PL invariant regular genus is additive under the connected sum within the class of all PL 4-manifolds admitting a weak semi-simple crystallization. Also, we note that this property is related to the 4-dimensional Smooth Poincar\'e Conjecture.

Characterization of finite groups generated by reflections and rotations

We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.

Lower bounds for regular genus and gem-complexity of PL 4-manifolds

Abstract Within crystallization theory, two interesting PL invariants for d-manifolds have been introduced and studied, namely, gem-complexity and regular genus. In the present paper we prove that for any closed connected PL 4-manifold M, its gem-complexity k ⁢ ( M ) {k(M)} and its regular genus 𝒢 ⁢ ( M ) {\mathcal{G}(M)} satisfy k ⁢ ( M ) ≥ 3 ⁢ χ ⁢ ( M ) + 10 ⁢ m - 6 and 𝒢 ⁢ ( M ) ≥ 2 ⁢ χ ⁢ ( M ) + 5 ⁢ m - 4 , k(M)\geq 3\chi(M)+10m-6\quad\text{and}\quad\mathcal{G}(M)\geq 2\chi(M)+5m-4, where r ⁢ k ⁢ ( π 1 ⁢ ( M ) ) = m {rk(\pi_{1}(M))=m} . These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of product 4-manifolds. Moreover, the class of semi-simple crystallizations is introduced, so that the represented PL 4-manifolds attain the above lower bounds. The additivity of both gem-complexity and regular genus with respect to connected sum is also proved for such a class of PL 4-manifolds, which comprehends all ones of “standard type”, involved in existing crystallization catalogs, and their connected sums.

Monodromy representations of completed coverings

In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterization of the discreteness of a completed normal covering. We also show that a completed covering admitting a monodromy representation is discrete and that the image of the branch set is closed.

Reflection positivity in simplicial gravity

Within the context of piecewise linear manifolds we establish reflection positivity with a Hilbert action given in terms of the Regge curvature and a cosmological term. Using this positivity a Hilbert space for a quantum theory is constructed and some field operators and observables are given. The set-up allows to introduce time reversal though no time exists. All constructions are non-perturbative.

Piecewise linear manifolds: Einstein metrics and Ricci flows**Dedicated to Ludwig Faddeev on the occasion of his 80th birthday.

This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear (p.l.) spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field. On a given set of p.l. spaces we define and discuss (normalized) Einstein flows. p.l. Einstein metrics are defined and examples are provided. Criteria for flows to approach Einstein metrics are formulated. Second variations of the total scalar curvature at a specific Einstein space are calculated.

Sparse Penalized Forward Selection for Support Vector Classification

We propose a new binary classification and variable selection technique especially designed for high-dimensional predictors. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection along with classification. By adding an ℓ1-type penalty to the loss function, common classification methods such as logistic regression or support vector machines (SVM) can perform variable selection. Existing penalized SVM methods all attempt to jointly solve all the parameters involved in the penalization problem altogether. When data dimension is very high, the joint optimization problem is very complex and involves a lot of memory allocation. In this article, we propose a new penalized forward search technique that can reduce high-dimensional optimization problems to one-dimensional optimization by iterating the selection steps. The new algorithm can be regarded as a forward selection version of the penalized SVM and its variants. The advantage of optimizing in one dimension is that the location of the optimum solution can be obtained with intelligent search by exploiting convexity and a piecewise linear or quadratic structure of the criterion function. In each step, the predictor that is most able to predict the outcome is chosen in the model. The search is then repeatedly used in an iterative fashion until convergence occurs. Comparison of our new classification rule with ℓ1-SVM and other common methods show very promising performance, in that the proposed method leads to much leaner models without compromising misclassification rates, particularly for high-dimensional predictors.

Certified computation of planar Morse–Smale complexes

The Morse–Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse–Smale complexes have been presented in case of piecewise linear manifolds (Edelsbrunner et al., 2003a). However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse–Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse–Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse–Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse–Smale complexes.

Simple crystallizations of 4-manifolds

Abstract Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of “standard” type, that is, all connected sums of ℂℙ2, S2 × S2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of ℂℙ2 is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of ℂℙ2 by the minimum of four edge contractions.

Cataloguing PL 4-Manifolds by Gem-Complexity

We describe an algorithm to subdivide automatically a given set of PL $n$-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n=4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices).Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.

Piecewise-linear manifold learning: A heuristic approach to non-linear dimensionality reduction

The need to reduce the dimensionality of a dataset whilst retaining its inherent manifold structure is key to many pattern recognition, machine learning, and computer vision problems. This process is often referred to as manifold learning since the structure is preserved during dimensionality reduction by learning the intrinsic low-dimensional manifold that the data lies upon. In this paper a heuristic approach is presented to tackle this problem by approximating the manifold as a set of piecewise linear models. By merging these linear models in an order defined by their global topology a globally stable and locally accurate model of the manifold can be obtained. A detailed analysis of the proposed approach is presented along with comparison with existing manifold learning techniques. Results obtained on both artificial and image based data show that in many cases this heuristic approach to manifold learning is able to out-perform traditional techniques.

Finite group actions on Kervaire manifolds

Let MK4k+2 be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1×S2k+1 of spheres. We show that a finite group of odd order acts freely on MK4k+2 if and only if it acts freely on S2k+1×S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k≠2j−1, then MK4k+2 does not admit any free TOP involutions. Free “exotic” (PL) involutions are constructed on MK30, MK62, and MK126. Each smooth structure on MK30 admits a free Z/2×Z/2 action.

Hysteretic autoregressive time series models

This paper extends the classical two-regime threshold autoregressive model by introducing hysteresis to its regime-switching structure, which leads to a new model: the hysteretic autoregressive model. The proposed model enjoys the piecewise linear structure of a threshold model but has a more flexible regime switching mechanism. A sufficient condition is given for geometric ergodicity. Conditional least squares estimation is discussed, and the asymptotic distributions of its estimators and information criteria for model selection are derived. Simulation results and an example support the model.

Norm minima in certain Siegel leaves

In this paper we shall illustrate that each polytopal moment-angle complex can be understood as the intersection of the minima of corresponding Siegel leaves and the unit sphere, with respect to the maximum norm. Consequently, an alternative proof of a rigidity theorem of Bosio and Meersseman is obtained; as piecewise linear manifolds, polytopal real moment-angle complexes can be smoothed in a natural way.

Acoustic space learning for sound-source separation and localization on binaural manifolds.

In this paper, we address the problems of modeling the acoustic space generated by a full-spectrum sound source and using the learned model for the localization and separation of multiple sources that simultaneously emit sparse-spectrum sounds. We lay theoretical and methodological grounds in order to introduce the binaural manifold paradigm. We perform an in-depth study of the latent low-dimensional structure of the high-dimensional interaural spectral data, based on a corpus recorded with a human-like audiomotor robot head. A nonlinear dimensionality reduction technique is used to show that these data lie on a two-dimensional (2D) smooth manifold parameterized by the motor states of the listener, or equivalently, the sound-source directions. We propose a probabilistic piecewise affine mapping model (PPAM) specifically designed to deal with high-dimensional data exhibiting an intrinsic piecewise linear structure. We derive a closed-form expectation-maximization (EM) procedure for estimating the model parameters, followed by Bayes inversion for obtaining the full posterior density function of a sound-source direction. We extend this solution to deal with missing data and redundancy in real-world spectrograms, and hence for 2D localization of natural sound sources such as speech. We further generalize the model to the challenging case of multiple sound sources and we propose a variational EM framework. The associated algorithm, referred to as variational EM for source separation and localization (VESSL) yields a Bayesian estimation of the 2D locations and time-frequency masks of all the sources. Comparisons of the proposed approach with several existing methods reveal that the combination of acoustic-space learning with Bayesian inference enables our method to outperform state-of-the-art methods.

Optimal lot-sizing and joint replenishment strategy under a piecewise linear warehousing cost structure

In the general lot-sizing replenishment problem, warehousing costs are considered as part of the inventory holding cost. When decision makers need warehouse space to store their products, they generally think about the warehousing cost as included in the inventory cost. In this study, we investigate an optimal lot-sizing and replenishment problem under a piecewise linear warehousing cost structure. Theoretical analysis of the optimality structure of a model of multiple stock-keeping units with joint inventory costs is conducted, and an effective search algorithm is proposed. Based on our random experiments, we demonstrate that our search algorithm outperforms the approach that the existing literature offers as solution.