| 1. |
- Adiprasito, Karim, et al.
(författare)
-
Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals
- 2017
-
Ingår i: Journal of the European Mathematical Society (Print). - : European Mathematical Society Publishing House. - 1435-9855 .- 1435-9863. ; 19:12, s. 3851-3865
-
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.
|
|
| 2. |
- Billera, L. J., et al.
(författare)
-
Face numbers of polytopes and complexes
- 2017
-
Ingår i: Handbook of Discrete and Computational Geometry, Third Edition. - : CRC Press. - 9781498711425 - 9781498711395 ; , s. 449-475
-
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.
|
|
| 3. |
- Björnberg, Jakob Erik, 1983-
(författare)
-
Graphical representations of Ising and Potts models : Stochastic geometry of the quantum Ising model and the space-time Potts model
- 2009
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in . In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in and in ‘star-like’ geometries.
|
|
| 4. |
- Björner, Anders
(författare)
-
A cell complex in number theory
- 2011
-
Ingår i: Advances in Applied Mathematics. - : Elsevier BV. - 0196-8858 .- 1090-2074. ; 46:1-4, s. 71-85
-
Tidskriftsartikel (refereegranskat)abstract
- Let Delta(n) be the simplicial complex of squarefree positive integers less than or equal to n ordered by divisibility. It is known that the asymptotic rate of growth of its Euler characteristic (the Mertens function) is closely related to deep properties of the prime number system. In this paper we study the asymptotic behavior of the individual Betti numbers beta(k)(Delta(n)) and of their sum. We show that Delta(n) has the homotopy type of a wedge of spheres, and that as n -> infinity S beta(k)(Delta(n)) = 2n/pi(2) + O(n(theta)), for all theta > 17/54, Furthermore, for fixed k, beta k(Delta(n)) similar to n/2logn (log log n)(k)/k!. As a number-theoretic byproduct we obtain inequalities partial derivative(k)(sigma(odd)(k+1)(n)) infinity S beta k((Delta) over tilde (n)) = n/3 + O(n(theta)), for all theta > 22/27.
|
|
| 5. |
|
|
| 6. |
- Björner, Anders
(författare)
-
A comparison theorem for f-vectors of simplicial polytopes
- 2007
-
Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 3:1, s. 347-356
-
Tidskriftsartikel (refereegranskat)abstract
- Let f(i)(P) denote the number of i-dimensional faces of a convex polytope P. Furthermore, let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Our main result is that for every simplicial d-polytope P, if f(r) (S (n(1), d)) <= f(r) (P) <= f(r) (C (n(2), d)) for some integers n(1), n(2) and r, then f(s) (S (n(1), d)) <= f(s) (P) <= f(s) (C (n(2), d)) for all s such that r < s. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain comparison theorem for f-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.
|
|
| 7. |
- Björner, Anders, et al.
(författare)
-
A note on blockers in posets
- 2004
-
Ingår i: Annals of Combinatorics. - : Springer Science and Business Media LLC. - 0218-0006 .- 0219-3094. ; 8:2, s. 123-131
-
Tidskriftsartikel (refereegranskat)abstract
- The blocker A* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: (1) The posets P for which A** = A for all antichains are characterized.(2) The blocker A* of a symmetric antichain in the partition lattice is characterized.(3) Connections with the question of finding minimal size blocking sets for certain set families are discussed.
|
|
| 8. |
- Björner, Anders
(författare)
-
A {$q$}-analogue of the {FKG} inequality and some applications
- 2011
-
Ingår i: Combinatorica. - : Springer Science and Business Media LLC. - 0209-9683 .- 1439-6912. ; 31:2, s. 151-164
-
Tidskriftsartikel (refereegranskat)abstract
- Let L be a finite distributive lattice and mu: L -> R(+) a log-supermodular function k: L -> R(+) let [GRAPHICS] We prove for any pair g, h: L -> R(+) of monotonely increasing functions, that E mu(g; q) . E mu(h; q) << E mu(1; q) . E mu(gh; q), where "<<" denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to q=1. The polynomial FKG inequality has applications to f-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of Schubert varieties, and to correlation-type inequalities for a class of power series weighted by Young tableaux. This class contains series involving Plancherel measure for the symmetric groups and its poissonization.
|
|
| 9. |
- Björner, Anders., et al.
(författare)
-
Bier spheres and posets
- 2005
-
Ingår i: Discrete & Computational Geometry. - : Springer Science and Business Media LLC. - 0179-5376 .- 1432-0444. ; 34:1, s. 71-86
-
Tidskriftsartikel (refereegranskat)abstract
- In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n -2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that cut across an ideal. Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular generalized Bier spheres. In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields many shellable spheres, most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.
|
|
| 10. |
- Björner, Anders, et al.
(författare)
-
Chain polynomials of distributive lattices are 75% unimodal
- 2005
-
Ingår i: The Electronic Journal of Combinatorics. - : The Electronic Journal of Combinatorics. - 1097-1440 .- 1077-8926. ; 12:1
-
Tidskriftsartikel (refereegranskat)abstract
- It is shown that the numbers c(i) of chains of length i in the proper part L\{0, 1} of a distributive lattice L of length l + 2 satisfy the inequalities c(0) <...< c([l/2]) and c([3l.4]) >... > c(l). This proves 75% of the inequalities implied by the Neggers unimodality conjecture.
|
|
