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A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(E)$,
Externí odkaz:
http://arxiv.org/abs/2410.21651
Autor:
De Loera, Javier
For cluster algebras of surface type, Musiker, Schiffler and Williams gave a formula for cluster variables in terms of perfect matchings of snake graphs. Building on this, we provide a simple determinantal formula for cluster variables via the weight
Externí odkaz:
http://arxiv.org/abs/2410.14554
We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of
Externí odkaz:
http://arxiv.org/abs/2403.09927
We study the problem of counting lattice points of a polytope that are weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials, as well as obt
Externí odkaz:
http://arxiv.org/abs/2402.11328
We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statis
Externí odkaz:
http://arxiv.org/abs/2402.09568
In this paper, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the fundamental theorem for
Externí odkaz:
http://arxiv.org/abs/2306.06270
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of possible
Externí odkaz:
http://arxiv.org/abs/2304.14239
Autor:
Bajo, Esme, Davis, Robert, De Loera, Jesús A., Garber, Alexey, Mora, Sofía Garzón, Jochemko, Katharina, Yu, Josephine
Publikováno v:
Adv. in Math.. {\bf 444} (2024), 109627
We generalize R. P. Stanley's celebrated theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for we
Externí odkaz:
http://arxiv.org/abs/2303.09614
In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Caratheodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the
Externí odkaz:
http://arxiv.org/abs/2212.02452
Publikováno v:
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation (ISSAC 2022). 2022. 333-342
Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity of $\mathc
Externí odkaz:
http://arxiv.org/abs/2210.03262