scholarly journals Resolutions of Convex Geometries

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Domenico Cantone ◽  
Jean-Paul Doignon ◽  
Alfio Giarlotta ◽  
Stephen Watson
Keyword(s):  
Convex Geometry ◽  
Learning Spaces ◽  
Combinatorial Structures ◽  
Open Problems ◽  
Set Systems ◽  
Convex Geometries

Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces.  We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry.  Contrary to what happens for similar constructions–compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Möhring and Radermacher)–, resolutions of convex geometries always yield a convex geometry.   We investigate resolutions of special convex geometries: ordinal and affine.  A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine.  A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones.  We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements.  Several open problems are listed. 

scholarly journals Pruning processes and a new characterization of convex geometries

2009 ◽  
Vol 309 (10) ◽  
pp. 3083-3091 ◽  
Author(s):  
Federico Ardila ◽  
Elitza Maneva
Keyword(s):  
Convex Geometries

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals Secret Sharing Schemes on Sparse Homogeneous Access Structures with Rank Three

10.37236/1825 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jaume Martí-Farré ◽  
Carles Padró
Keyword(s):  
Secret Sharing ◽  
Access Structure ◽  
Open Problems ◽  
Access Structures ◽  
The Family ◽  
Ideal Access Structures ◽  
Optimal Information ◽  
Made In ◽  
The Ideal

One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most $2/3$. An access structure is said to be $r$-homogeneous if there are exactly $r$ participants in every minimal qualified subset. A first approach to the characterization of the ideal $3$-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse $3$-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse $3$-homogeneous access structures.

A CHARACTERIZATION OF RECOGNIZABLE PICTURE LANGUAGES

1994 ◽  
Vol 08 (02) ◽  
pp. 501-508 ◽  
Author(s):  
KATSUSHI INOUE ◽  
ITSUO TAKANAMI
Keyword(s):  
Finite Automata ◽  
Two Dimensional ◽  
Open Problems ◽  
Nondeterministic Finite Automata ◽  
Letter Alphabet ◽  
The Family ◽  
On Line ◽  
Picture Languages

This paper first shows that REC, the family of recognizable picture languages in Giammarresi and Restivo,3 is equal to the family of picture languages accepted by two-dimensional on-line tessellation acceptors in Inoue and Nakamura.5 By using this result, we then solve open problems in Giammarresi and Restivo,3 and show that (i) REC is not closed under complementation, and (ii) REC properly contains the family of picture languages accepted by two-dimensional nondeterministic finite automata even over a one letter alphabet.

scholarly journals On(k,kn-k2-2k-1)-Choosability ofn-Vertex Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Wongsakorn Charoenpanitseri
Keyword(s):  
Proper Coloring ◽  
Open Problems

A(k,t)-list assignmentLof a graphGis a mapping which assigns a set of sizekto each vertexvofGand|⋃v∈V(G)‍L(v)|=t. A graphGis(k,t)-choosable ifGhas a proper coloringfsuch thatf(v)∈L(v)for each(k,t)-list assignmentL. In 2011, Charoenpanitseri et al. gave a characterization of(k,t)-choosability ofn-vertex graphs whent≥kn-k2-2k+1and left open problems whent≤kn-k2-2k. Recently, Ruksasakchai and Nakprasit obtain the results whent=kn-k2-2k. In this paper, we extend the results to caset=kn-k2-2k-1.

scholarly journals AE Regularity of Interval Matrices

2018 ◽  
Vol 33 ◽  
pp. 137-146
Author(s):  
Milan Hladík
Keyword(s):  
Computational Complexity ◽  
Coefficient Matrix ◽  
Solution Set ◽  
System Of Equations ◽  
Existential Quantifier ◽  
Open Problems ◽  
Linear System Of Equations ◽  
Classes Of Matrices ◽  
Interval Coefficients

Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by ∀∃- quantification. The paper deals with the problem of what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, a concept of AE regularity is introduced, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. A characterization of AE regularity is discussed, and also various classes of matrices that are implicitly AE regular are investigated. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. Eventually, there are also stated open problems related to computational complexity and characterization of AE regularity.

scholarly journals The Prime Stems of Rooted Circuits of Closure Spaces and Minimum Implicational Bases

10.37236/3068 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Masataka Nakamura ◽  
Kenji Kashiwabara
Keyword(s):  
Convex Geometry ◽  
Closure Operators ◽  
Optimal Basis ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Systems ◽  
Closure Spaces ◽  
Convex Geometries ◽  
Necessary And Sufficient ◽  
Special Case

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair $(X,e)$ of a subset $X$, called a stem, and an element $e\not\in X$, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore  gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessarily sufficient to restore the original antimatroid as an implicational system. It is shown through an example.

scholarly journals NBC Complexes of Convex Geometries

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Kenji Kashiwabara ◽  
Masataka Nakamura
Keyword(s):  
Convex Geometry ◽  
Decomposition Theorem ◽  
International Audience ◽  
Generating Theorem ◽  
Convex Geometries ◽  
Broken Circuit

International audience We introduce a notion of a $\textit{broken circuit}$ and an $\textit{NBC complex}$ for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota's formula and Brylawski's decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem.

scholarly journals Toric varieties and Gröbner bases: the complete $$\mathbb {Q}$$-factorial case

2020 ◽  
Vol 31 (5-6) ◽  
pp. 461-482
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Toric Varieties ◽  
Toric Ideals ◽  
Projective Embedding ◽  
Complete Case ◽  
Open Problems ◽  
Toric Ideal ◽  
Definition Of ◽  
Nef Cone ◽  
Special Fibre

Abstract We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by V and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels’ arguments on the Gröbner fan of toric ideals to our complete case; we give a characterization of the Gröbner region and show an explicit correspondence between Gröbner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to V allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by V. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $$\mathbb {Q}$$ Q -factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.

scholarly journals The continuous Diophantine approximation mapping of Szekeres

1995 ◽  
Vol 59 (2) ◽  
pp. 148-172 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Andrew D. Pollington
Keyword(s):  
Diophantine Approximation ◽  
Continued Fraction ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Unit Square ◽  
Alternative Characterization ◽  
Continuous Analogue

AbstractSzekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

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