| 1. |
- Brattström, Gudrun
(författare)
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Leonardo da Pisa : Inger Christensen och Fibonacci
- 2008
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Ingår i: Lyrikvännen. - 0460-0762. ; 55:6, s. 33-38
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Tidskriftsartikel (populärvet., debatt m.m.)abstract
- Den danska poeten Inger Christensen byggde sin diktsamling Alfabet på Fibonaccis talserie. Artikeln är en kort historisk essä över Fibonacci, och beskriver även hur talserien är konstruerad.
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| 2. |
- Ekedahl, Torsten, 1955-
(författare)
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Shellability of a Poset of Polygonal Subdivisions
- 2009
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Ingår i: The Electronic Journal of Combinatorics. - : Annals of Mathematics. - 1097-1440 .- 1077-8926. ; 16:2, s. R23-
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Tidskriftsartikel (refereegranskat)
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| 3. |
- Krüger, Oliver, 1988-
(författare)
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On linear graph invariants related to Ramsey and edge numbers : or how I learned to stop worrying and love the alien invasion
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we study the Ramsey numbers, R(l,k), the edge numbers, e(l,k;n) and graphs that are related to these. The edge number e(l,k;n) may be defined as the least natural number m for which all graphs on n vertices and less than m edges either contains a complete subgraph of size l or an independent set of size k. The Ramsey number R(l,k) may then be defined as the least natural number n for which e(l,k;n) = ∞ .In Paper I, IV and V we study strict lower bounds for e(l,k;n). In Paper I we do this in the case where l = 3 by, in particular, showing e(G) ≥ (1/3)(17n(G) - 35α(G) - N(C4;G)) for all triangle-free graphs G, where N(C4;G) denotes the number of cycles of length 4 in G. In Paper IV we describe a general method for generating similar inequalities to the one above but for graphs that may contain triangles, but no complete subgraphs of size 4. We then show a selection of the inequalities we get from the computerised generation. In Paper V we study the inequality e(G) ≥ (1/2)(ceil((7l - 2)/2)n(G) - l floor((5l + 1)/2)α(G))for l ≥ 2, and examine the properties of graphs G without cliques of size l+1 such that G is minimal with respect to the above inequality not holding, and show for small l that no such graphs G can exist.In Paper II we study constructions of graphs G such that e(G) - e(3,k;n) is small when n ≤ 3.5(k-1). We employ a description of some of these graphs in terms of 'patterns' and a recursive procedure to construct them from the patterns. We also present the result of computer calculations where we actually have performed such constructions of Ramsey graphs and compare these lists to previously computed lists of Ramsey graphs.In Paper III we develop a method for computing, recursively, upper bounds for Ramsey numbers R(l,k). In particular the method uses bounds for the edge numbers e(l,k;n). In Paper III we have implemented this method as a computer program which we have used to improve several of the best known upper bounds for small Ramsey numbers R(l,k).
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| 4. |
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| 5. |
- Strohmayer, Henrik
(författare)
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Operads of compatible structures and weighted partitions
- 2008
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Ingår i: Journal of Pure and Applied Algebra. - : Elsevier BV. - 0022-4049. ; 212:11, s. 2522-
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and pre-Lie algebras.
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| 6. |
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| 7. |
- Berglund, Alexander
(författare)
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Shellability and the strong gcd-condition
- 2009
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Ingår i: The Electronic Journal of Combinatorics. - 1097-1440 .- 1077-8926. ; 16:2
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Tidskriftsartikel (refereegranskat)abstract
- Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.
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| 8. |
- Crispin Quiñonez, Veronica, et al.
(författare)
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On ideals generated by two generic quadratic forms in the exterior algebra
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.
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| 9. |
- Deijfen, Maria, et al.
(författare)
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Competing frogs on Z
- 2019
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Ingår i: Electronic Journal of Probability. - 1083-6489 .- 1083-6489. ; 24
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Tidskriftsartikel (refereegranskat)abstract
- A two-type version of the frog model on Z^d is formulated, where active type i particles move according to lazy random walks with probability p_i of jumping in each time step (i = 1; 2). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles atone other site are activated and assigned type 2, while all other particles are sleeping. When an active type i particle moves to a new site, any sleeping particles there are activated and assigned type i, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. Let G_i denote the event that type i activates infinitely many particles. We show that the events G_1 intersect G_2^c and G_1^c intersect G_2 both have positive probability for all p_1,p_2 in (0,1]. Furthermore, if p_1 = p_2, then the types can coexist in the sense that the event G_1 intersect G_2 has positive probability. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when p_1 not equal p_2.
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| 10. |
- Nenashev, Gleb, 1992-
(författare)
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Around power ideals : From Fröberg's conjecture to zonotopal algebra
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we study power algebras, which are quotient of polynomial rings by power ideals. We will study Hilbert series of such ideals and their other properties. We consider two important special cases, namely, zonotopal ideals and generic ideals. Such ideals have a lot combinatorial properties.In the first chapter we study zonotopal ideals, which were defined and used in several earlier publications. The most important works are by F.Ardila and A.Postnikov and by O.Holtz and A.Ron. These papers originate from different sources, the first source is homology theory, the second one is the theory of box splines. We study quotient algebras by these ideals; these algebras have a nice interpretation for their Hilbert series, as specializations of their Tutte polynomials. There are two important subclasses of these algebras, called unimodular and graphical. The graphical algebras were defined by A.Postnikov and B.Shapiro. In particular, the external algebra of a complete graph is exactly the algebra generated by the Bott-Chern forms of the corresponding complete flag variety. One of the main results of the thesis is a characterization of external algebras. In fact, for the case of graphical and unimodular algebras we prove that external algebras are in one-to-one correspondence with graphical and regular matroids, respectively.In the second chapter we study Hilbert series of generic ideals. By a generic ideal we mean an ideal generated by forms from some class, whose coefficients belong to a Zariski-open set. There are two main classes to consider: the first class is when we fix the degrees of generators; the famous Fröberg's conjecture gives the expected Hilbert series of such ideals; the second class is when an ideal is generated by powers of generic linear forms. There are a few partial results on Fröberg's conjecture, namely, when the number of variables is at most three. In both classes the Hilbert series is known in the case when the number of generators is at most (n+1). In both cases we construct a lot of examples when the degree of generators are the same and the Hilbert series is the expected one.
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