165,041 research outputs found
Borsuk and V\'azsonyi problems through Reuleaux polyhedra
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 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
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
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
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
Let be a differentially closed field of
characteristic with field of constants .
In the first part of the paper we explore the connection between Ax-Schanuel
type theorems (predimension inequalities) for a differential equation
and the geometry of the fibres where
is a non-constant element. We show that certain types of predimension
inequalities imply strong minimality and geometric triviality of .
Moreover, the induced structure on the Cartesian powers of is given by
special subvarieties. In particular, since the -function satisfies an
Ax-Schanuel inequality of the required form (due to Pila and Tsimerman),
applying our results to the -function we recover a theorem of Freitag and
Scanlon stating that the differential equation of defines a strongly
minimal set with trivial geometry.
In the second part of the paper we study strongly minimal sets in the
-reducts of differentially closed fields. Let be the
(two-variable) differential equation of the -function. We prove a Zilber
style classification result for strongly minimal sets in the reduct
. More precisely, we show that in
all strongly minimal sets are geometrically trivial or non-orthogonal to .
Our proof is based on the Ax-Schanuel theorem and a matching Existential
Closedness statement which asserts that systems of equations in terms of
have solutions in 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
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
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.
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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