arXiv subjects
Publications and source records attributed to Gabor Hegedüs.
R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.
P. Frankl and J. Pach proved the following uniform version of Sauer's Lemma. Let $n,d,s$ be natural numbers such that $d\leq n$, $s+1\leq n/2$. Let $\cF \subseteq {[n] \choose d}$ be an arbitrary $d$-uniform set system such that $\cF$ does not shatter an $s+1$-element set, then $$ |\cF|\leq {n \choose s}.$$ We prove here two generalizations of the above theorem to $n$-tuple systems. To obtain these results, we use Gröbner basis methods, and describe the standard monomials of Hamming spheres.
Let $Δ$ be simplicial complex and let $k[Δ]$ denote the Stanley--Reisner ring corresponding to $Δ$. Suppose that $k[Δ]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity of $k[Δ]$ in terms of the $h$--vector of $Δ$. As an application, we derive a linear equation system and some inequalities for the components of the $h$--vector of the clique complex of an arbitrary chordal graph. As an other application, we derive a linear equation system and some inequalities for the components of the $h$--vector of Cohen--Macaulay simplicial complexes.
Let $Δ$ be simplicial complex and let $k[Δ]$ denote the Stanley--Reisner ring corresponding to $Δ$. Suppose that $k[Δ]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity of $k[Δ]$ in terms of the $h$--vector of $Δ$. As an application, we derive a linear equation system for the components of the $h$--vector of the clique complex of an arbitrary chordal graph.
We describe the Betti numbers of the edge ideals $I(G)$ of uniform hypergraphs $G$ such that $I(G)$ has linear graded free resolution. We give an algebraic equation system and some inequalities for the components of the $f$--vector of the clique complex of an arbitrary chordal graph. Finally we present an explicit formula for the multiplicity of the Stanley-Reisner ring of the edge ideals of any chordal graph.
We characterize and classify completely the planar 4R closed chain working on the Minkowskian plane. Our work would open a new research direction in the theory of geometric designs: the classification and characterization of the geometric design of linkages working in non--Euclidean spaces.