165,041 research outputs found

    Borsuk and V\'azsonyi problems through Reuleaux polyhedra

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    The Borsuk conjecture and the V\'azsonyi problem are two attractive and famous questions in discrete and combinatorial geometry, both based on the notion of diameter of a bounded sets. In this paper, we present an equivalence between the critical sets with Borsuk number 4 in R3\mathbb{R}^3 and the minimal structures for the V\'azsonyi problem by using the well-known Reuleaux polyhedra. The latter lead to a full characterization of all finite sets in R3\mathbb{R}^3 with Borsuk number 4. The proof of such equivalence needs various ingredients, in particular, we proved a conjecture dealing with strongly critical configuration for the V\'azsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical

    On the complexity of strongly connected components in directed hypergraphs

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    We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure

    The J-minimal sets in the hereditary theories

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    Our attention in given article will be paid to the study of model - theoretic properties of hereditary Jonsson theories, while we consider Jonsson theories that retain jonsonness under any admissible enrichment. In given paper new concepts of ¾essential type¿, ¾essential geometric base¿ are introduced, the orbital types and strongly minimal sets within the framework of special subsets of the semantic model, on which a closure operator is given, defining the special geometry of Jonsson are considered. The results for the J-strongly minimal types of the semantic model in the case, when these sets are separated from the orbits of the central types of Jonsson hereditary theories are also obtained

    Ax-Schanuel and strong minimality for the j-function

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    Let K:=(K;+,⋅,D,0,1)\mathcal{K}:=(K;+,\cdot, D, 0, 1) be a differentially closed field of characteristic 00 with field of constants CC. In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x,y)E(x,y) and the geometry of the fibres Us:={y:E(s,y)∧y∉C}U_s:=\{ y:E(s,y) \wedge y \notin C \} where ss is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of UsU_s. Moreover, the induced structure on the Cartesian powers of UsU_s is given by special subvarieties. In particular, since the jj-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the jj-function we recover a theorem of Freitag and Scanlon stating that the differential equation of jj defines a strongly minimal set with trivial geometry. In the second part of the paper we study strongly minimal sets in the jj-reducts of differentially closed fields. Let Ej(x,y)E_j(x,y) be the (two-variable) differential equation of the jj-function. We prove a Zilber style classification result for strongly minimal sets in the reduct K:=(K;+,⋅,Ej)\mathsf{K}:=(K;+, \cdot, E_j). More precisely, we show that in K\mathsf{K} all strongly minimal sets are geometrically trivial or non-orthogonal to CC. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of EjE_j have solutions in K\mathsf{K} unless having a solution contradicts Ax-Schanuel.Comment: 27 pages. This is a combination of arXiv:1606.01778v3 and arXiv:1805.03985v1 (with substantial revisions

    Pregeometry on the subsets of Jonsson theory’s semantic model

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    One of the interesting achievements among investigations from modern model theory is the implement of local properties of the geometry of strongly minimal sets. In E. Hrushovski have proved remarkable results under this ideas and this one had impacted an essential infuence for development of methods and ideas of research for global properties of structures.These new model - theoretical features and approvals play an important role in E. Hrushovski’s proof of the Mordell-Lang Conjecture for function fields. In this article, we are trying to redefine the basic concepts of the above mentioned ideas on the formul subsets of some extentional - closed model for some fixed Jonsson theory. With the help of new concepts in the frame of Jonssoness features, pregeometry is given on all subsets of Jonsson theory’s semantic model. Minimal structures and, correspondingly, pregeometry and geometry of minimal structures are determined. We consider the concepts of dimension, independence, and basis in the Jonsson strongly minimal structures for Jonsson theories

    Zariski Geometries

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    We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.Comment: 9 page

    The geometry of Hrushovski constructions, II. The strongly minimal case.

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    We investigate the isomorphism types of combinatorial geometries arising from Hrushovski's at strongly minimal structures and answer some questions from Hrushovski's original paper
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