A simplicial 3-arrangement of 22 planes

This paper describes a join-irreducible simplicial arrangement A _inf1 ^sup3 (22) of 22 planes in real projective 3-space that is not included in the list recently published by Branko Grünbaum and G. C. Shephard. Seventeen sporadic 3-arrangements are now known.     

A generalization of simplicial elimination orderings

We consider the class of graphs where every induced subgraph possesses a vertex whose neighborhood has no P₄ and no 2K₂. We prove that Berge's Strong Perfect Graph Conjecture holds for such graphs. The class generalizes several well-known families of perfect graphs, such as triangulated graphs and bipartite graphs. Testing membership in this class and computing the maximum clique size for a graph in this class is not hard, but finding an optimal coloring is NP-hard. We give a polynomial-time algorithm for optimally coloring the vertices of such a graph when it is perfect. © 1996 John Wiley & Sons, Inc.     

A combinatorial decomposition of simplicial complexes

We find a decomposition of simplicial complexes that implies and sharpens the characterization (due to Björner and Kalai) of thef-vector and Betti numbers of a simplicial complex. It generalizes a result of Stanley, who proved the acyclic case, and settles a conjecture of Stanley and Kalai.     

A combinatorial decomposition of acyclic simplicial complexes

It is proved that if Δ is a finite acyclic simplicial complex, then there is a subcomplex Δ′ Δ and a bijection η: Δ′ → Δ − Δ′ such that F η(F) and |η(F)−F|=1 for all F Δ′. This improves an earlier result of Kalai. An immediate corollary is a characterization (first due to Kalai) of the f-vector of an acyclic simplicial complex. Several generalizations, some proved and some conjectured, are discussed.     

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n,K), then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ(8,K) and show that every locally singular hyperplane of DQ(8,K) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.     

Arrangements of oriented hyperplanes

An arrangement ofn oriented hyperplanes or half-spaces dividesE ^d into a certain number of convex cells. We study the numberc _k of cells which are covered by exactlyk half-spaces and derive an upper bound onc _k for givenn andd.     

An A ∞ structure on simplicial complexes

We consider a discrete (finite-difference) analogue of differential forms defined on simplicial complexes, in particular, on triangulations of smooth manifolds. Various operations are explicitly defined on these forms including the exterior differential d and the exterior product ∧. The exterior product is nonassociative but satisfies a more general relation, the so-called A _∞ structure. This structure includes an infinite set of operations constrained by the nilpotency relation (d + ∧ + m + …)ⁿ = 0 of the second degree, n = 2. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 3–37, July, 2008.     

A Gorenstein simplicial complex for symmetric minors

We show that the ideal generated by the (n - 2) minors of a general symmetric n by n matrix has an initial ideal that is the Stanley–Reisner ideal of the boundary complex of a simplicial polytope and has the same graded Betti numbers.     

A simplicial homology algorithm for Lipschitz optimisation

The simplicial homology global optimisation (SHGO) algorithm is a general purpose global optimisation algorithm based on applications of simplicial integral homology and combinatorial topology. SHGO approximates the homology groups of a complex built on a hypersurface homeomorphic to a complex on the objective function. This provides both approximations of locally convex subdomains in the search space through Sperner’s lemma and a useful visual tool for characterising and efficiently solving higher dimensional black and grey box optimisation problems. This complex is built up using sampling points within the feasible search space as vertices. The algorithm is specialised in finding all the local minima of an objective function with expensive function evaluations efficiently which is especially suitable to applications such as energy landscape exploration. SHGO was initially developed as an improvement on the topographical global optimisation (TGO) method. It is proven that the SHGO algorithm will always outperform TGO on function evaluations if the objective function is Lipschitz smooth. In this paper SHGO is applied to non-convex problems with linear and box constraints with bounds placed on the variables. Numerical experiments on linearly constrained test problems show that SHGO gives competitive results compared to TGO and the recently developed Lc-DISIMPL algorithm as well as the PSwarm, LGO and DIRECT-L1 algorithms. Furthermore SHGO is compared with the TGO, basinhopping (BH) and differential evolution (DE) global optimisation algorithms over a large selection of black-box problems with bounds placed on the variables from the SciPy benchmarking test suite. A Python implementation of the SHGO and TGO algorithms published under a MIT license can be found from https://bitbucket.org/upiamcompthermo/shgo/.     

Non-regular simplicial matroids

Let (ks) denote the set of all k-element-subsets of a finite set S. A k-simplical matroid on a subset E of (ks) is a binary matroid the circuit of which are simplicial complexes {X1,…Xm} ? E with boundary 0 (mod 2). The k-simplical matroid on (ks) is called the full simplicial matroid Gk(S). The polygon matroid on the edges of a finite graph is 2-simplicial. Polygon-matroids and their duals are regular. The dual of Gk(S) is Gn?k(S) if the cardinnlity of S is n. More details on simplicial matroids can be found in [3, Chapter 6] and also in [4, pp. 180–181].Welsh asked if every simplicial matroid is regular. We prove that this is not the case, for all full k-simplicial matroids Gk(S) with 3?k?n?3 are non-regular (n is the cardinality of S). This result has also been proved σy R. Cordovil and M. Las Vergnas recently. Their proof is different from our proof, which is somewhat shorter.     

A simplicial complex of 2-stack sortable permutations

For each n, we construct a simplicial complex whose k-dimensional faces are in one-to-one correspondence with 2-stack sortable permutations of length n having k ascents.     

Homotopy simplicial faces and the homology of realizations of simplicial topological spaces

The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of A _∞-algebras and D _∞-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.