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Šachové úlohy v kombinatorice

Lucie Chybová (2018)

Pokroky matematiky, fyziky a astronomie

Článek pojednává o matematických úlohách souvisejících se šachovnicí a šachovými figurami. Ze šachu však budeme potřebovat pouze pravidla pro pohyb figur po šachovnici. Postupně se zaměřujeme na jezdcovy procházky po obdélníkových šachovnicích a dále na tzv. nezávislost a dominanci figur a vztah mezi nimi na čtvercových šachovnicích. Ukážeme, že některé problémy lze řešit elegantněji, pokud je přeformulujeme v řeči teorie grafů.

Sagbi bases of Cox–Nagata rings

Bernd Sturmfels, Zhiqiang Xu (2010)

Journal of the European Mathematical Society

We degenerate Cox–Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev–Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n -space at n + 3 points, sagbi bases of Cox–Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D’Cruz–Iarrobino and Buczyńska–Wiśniewski....

Saturation numbers for linear forests P 6 + t P 2

Jingru Yan (2023)

Czechoslovak Mathematical Journal

A graph G is H -saturated if it contains no H as a subgraph, but does contain H after the addition of any edge in the complement of G . The saturation number, sat ( n , H ) , is the minimum number of edges of a graph in the set of all H -saturated graphs of order n . We determine the saturation number sat ( n , P 6 + t P 2 ) for n ≥ 10 3 t + 10 and characterize the extremal graphs for n > 10 3 t + 20 .

Saturation numbers for trees.

Faudree, Jill, Faudree, Ralph J., Gould, Ronald J., Jacobson, Michael S. (2009)

The Electronic Journal of Combinatorics [electronic only]

Scale-free percolation

Maria Deijfen, Remco van der Hofstad, Gerard Hooghiemstra (2013)

Annales de l'I.H.P. Probabilités et statistiques

We formulate and study a model for inhomogeneous long-range percolation on ℤ d . Each vertex x ∈ ℤ d is assigned a non-negative weight W x , where ( W x ) x ∈ ℤ d are i.i.d. random variables. Conditionally on the weights, and given two parameters α , λ g t ; 0 , the edges are independent and the probability that there is an edge between x and y is given by p x y = 1 - exp { - λ W x W y / | x - y | α } . The parameter λ is the percolation parameter, while α describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there...

Schur and Schubert polynomials as Thom polynomials-cohomology of moduli spaces

László Fehér, Richárd Rimányi (2003)

Open Mathematics

The theory of Schur and Schubert polynomials is revisited in this paper from the point of view of generalized Thom polynomials. When we apply a general method to compute Thom polynomials for this case we obtain a new definition for (double versions of) Schur and Schubert polynomials: they will be solutions of interpolation problems.

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