| 1. |
- Adiprasito, Karim, et al.
(författare)
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Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals
- 2017
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Ingår i: Journal of the European Mathematical Society (Print). - : European Mathematical Society Publishing House. - 1435-9855 .- 1435-9863. ; 19:12, s. 3851-3865
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Tidskriftsartikel (refereegranskat)abstract
- A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.
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| 2. |
- Billera, L. J., et al.
(författare)
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Face numbers of polytopes and complexes
- 2017
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Ingår i: Handbook of Discrete and Computational Geometry, Third Edition. - : CRC Press. - 9781498711425 - 9781498711395 ; , s. 449-475
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Geometric objects are often put together from simple pieces according to certain combinatorial rules. As such, they can be described as complexes with their constituent cells, which are usually polytopes and often simplices. Many constraints of a combinatorial and topological nature govern the incidence structure of cell complexes and are therefore relevant in the analysis of geometric objects. Since these incidence structures are in most cases too complicated to be well understood, it is worthwhile to focus on simpler invariants that still say something nontrivial about their combinatorial structure. The invariants to be discussed in this chapter are the f-vectors f = (f 0, f 1, …) $ f=(f_0, f_1, \dots) $, where f i $ f_i $ is the number of i-dimensional cells in the complex.
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| 3. |
- Björner, Anders
(författare)
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Continuous Matroids Revisited
- 2019
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Ingår i: Building Bridges II. - Berlin, Heidelberg : Springer Nature. ; , s. 17-28
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Konferensbidrag (refereegranskat)abstract
- Here we look back at some work done in the mid-1980s in collaboration with László Lovász. Our main concern at that time was to provide conditions that make it possible to pass to the limit of a class of finite matroids. With the current flurry of interest in limits of combinatorial objects, a review of such matroid limits seems timely. The characteristic property of a continuous matroid is the existence of a rank function taking as values the full real unit interval. Known examples of such rank functions include Lebesgue measure on the unit interval and the dimension function of certain von Neumann algebras. In both these cases the lattice property of modularity plays a crucial role. A more general concept, pseudomodularity, makes possible the construction of e.g. continuous field extensions (algebraic matroids) and continuous partition lattices (graphic matroids).
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| 4. |
- Björner, Anders, et al.
(författare)
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On codimension one embedding of simplicial complexes
- 2017
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Ingår i: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek. - Cham : Springer International Publishing. - 9783319444796 - 9783319444789 ; , s. 207-219
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- We study d-dimensional simplicial complexes that are PL embeddable in Rd+1. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic approach to deriving upper bounds for the number of top-dimensional faces of such complexes, particularly in low dimensions.
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| 5. |
- Björner, Anders, et al.
(författare)
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On the connectivity of manifold graphs
- 2015
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 143:10, s. 4123-4132
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Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.
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| 6. |
- Björner, Anders.
(författare)
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Positive sum systems
- 2015
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Ingår i: Springer INdAM Series. - Cham : Springer International Publishing. ; , s. 157-171
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Konferensbidrag (refereegranskat)abstract
- Let x1, x2, …, xn be real numbers summing to zero, and let p+ be the family of all subsets J ⊆ [n]:={1,2,⋯n}such that (Formula presented). Subset families arising in this way are the objects of study here. We prove that the order complex of P+, viewed as a poset under set containment, triangulates a shellable ball whose f-vector does not depend on the choice of x, and whose h-polynomial is the classical Eulerian polynomial. Then we study various components of the flag f-vector of P+ and derive some inequalities satisfied by them. It has been conjectured by Manickam, Miklós and Singhi in 1986 that (Formula presented) is a lower bound for the number of k-element subsets in P+, unless n/k is too small. We discuss some related results that arise from applying the order complex and flag f-vector point of view. Some remarks at the end include brief discussions of related extensions and questions. For instance, we mention positive sum set systems arising in matroids whose elements are weighted by real numbers.
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| 7. |
- Björner, Anders., et al.
(författare)
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Using brouwer’s fixed point theorem
- 2017
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Ingår i: A Journey through Discrete Mathematics: A Tribute to Jiri Matousek. - Cham : Springer International Publishing. - 9783319444796 - 9783319444789 ; , s. 221-271
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Brouwer’s fixed point theorem from 1911 is a basic result in topology- with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain their applications to the fascinating (and still not fully solved) evasiveness problem.
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| 8. |
- Goodarzi, Afshin
(författare)
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Topological and Shifting Theoretic Methods in Combinatorics and Algebra
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of six papers related to combinatorics and commutative algebra.In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the generic initial ideal.In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.
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