The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the literature an axiomatic characterization of the angle defect. We give a characterization of the angle defect for simplicial surfaces, and we show that variants of the...

Abstract We present an e cient algorithm for solving a linear system arising from the 1-Laplacian of a collapsible simplicial complex with a known collapsing sequence. When combined with a result of Chillingworth, our algorithm is applicable to convex simplicial complexes embedded in R. The running time of our algorithm is nearly-linear in the size of the complex and is logarithmic on its numer...

Provan and Billera defined the notion of weak k-decomposability for pure simplicial complexes in the hopes of bounding the diameter of convex polytopes. They showed the diameter of a weakly k-decomposable simplicial complex ã is bounded above by a polynomial function of the number of k-faces in ã and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of verti...

Constructibility is a combinatorial property of simplicial complexes. In general, it requires a great deal of time to decide whether a simplicial complex is constructible or not. In this paper, we consider sufficient conditions for nonconstructibility of simplicial 3-balls to investigate efficient algorithms for the decision problem. © 2007 Elsevier B.V. All rights reserved.

In this paper we will develop a theory for simplicial diffeomorphims, that is, diffeomorphims that preserve the incidence relations of a simplicial complex, and analyze alternative schemes to construct them with different properties. In combining piecewise linear functions on complexes with simplicial diffeomorphisms, we propose a new representation of curves and surfaces (and hypersurfaces, in...

Reactions of chloro bridged complexes with R-3-(3-pyridyl)alanine afford the chelate complexes LnM[NH2CH(CO2)CH2C5H4NH+]Cl− (LnM = Ph3P(Cl)Pd, (tol3P)(Cl)Pd, (Ph-pyridyl)2Ir, Cp∗(Cl)Rh, Cp∗(Cl)Ir, (p-Cymol)(Cl)Ru) with protonated pyridine substituents. An analogous Cp∗Rh complex with 3-(2-pyridyl)alaninate was also obtained. Addition of base (NaOMe) to these complexes gives dimeric and trimeric...

In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k-dimensional Laplacian for 1 ≤ k ≤ d. We study an example of random walks on simplicial complexes in the context of a semi-supervised learning problem. Specifically, we...

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel’s conjecture ...

Abstract Liouville’s theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two complexes considered discretely conformally equivalent if they combinatorially and the lengths of corresponding edges related by scale factors associated with vertices.

We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in polynomial time whether a simplicial compl...