Searching for perfection on hypergraphs

In this paper we study the family of oriented transitive 3-hypergraphs that arise from cyclic permutations and intervals in the circle, in order to search for the notion of perfection on hypergraphs.     

Degree sequence of oriented k-hypergraphs

In this paper, we define the degree of a vertex in oriented hypergraphs and give a necessary and sufficient condition for a nondecreasing sequence of integers to be a degree sequence of an oriented 3-hypergraphs.     

A family of nonreconstructible hypergraphs

A k-hypergraph is a hypergraph in which each edge contains k vertices. We describe the construction of an infinite family of finite, nonreconstructible 3-hypergraphs. We also indicate why the same techniques can likely be used to construct nonreconstructible k-hypergraphs for any k ⩾ 3.     

On countable homogeneous 3-hypergraphs

Summary We present some results on countable homogeneous 3-hypergraphs. In particular, we show that there is no unexpected homogeneous 3-hypergraph determined by a single constraint.The authors are grateful for support from NSERC (Canada) Grant A3040     

New Linear Codes Over \mathbb{F}_3 and \mathbb{F}_5 and Improvements on Bounds

One of the most important problems of coding theory is to constructcodes with best possible minimum distances. In this paper, we generalize the method introduced by [8] and obtain new codes which improve the best known minimum distance bounds of some linear codes. We have found a new linear ternary code and 8 new linear codes over with improved minimumdistances. First we introduce a generalized version of Gray map,then we give definition of quasi cyclic codes and introduce nearlyquasi cyclic codes. Next, we give the parameters of new codeswith their generator matrices. Finally, we have included twotables which give Hamming weight enumerators of these new codes.     

Extendability of Cyclic Orders

A cyclic order is a ternary relation that satisfies ternary transitivity and asymmetry conditions. Such a ternary relation is extendable if it is included in a complete cyclic order on the same ground set. Unlike the case of linear extensions of partial orders, a cyclic order need not be extendable. The extension problem for cyclic orders is to determine if a cyclic order is extendable. This problem is known to be NP-complete. We introduce a class of cyclic orders in which the extension problem can be solved in polynomial time. The class provides many explicit examples of nonextendable cyclic orders that were not previously known, including a nonextendable cyclic order on seven points. Let μ be the maximum cardinality of a ground set on which all cyclic orders are extendable. It has been shown that μ≤9. We prove that μ=6. This answers a question of Novák. In addition, we characterize the nonextendable cyclic orders on seven and eight points. Our results are intimately related to irreducible partially ordered set of order dimension three, and to fractional vertices of generalized transitive tournament polytopes. As by-products, we obtain a characterization of cyclically ordered sets of dimension two, and a new proof of a theorem of Dridi on small linear ordering polytopes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation of Metric Spaces by Partial Metric Spaces

Partial metrics are generalised metrics with non-zero self-distances. We slightly generalise Matthews' original definition of partial metrics, yielding a notion of weak partial metric. After considering weak partial metric spaces in general, we introduce a weak partial metric on the poset of formal balls of a metric space. This weak partial metric can be used to construct the completion of classical metric spaces from the domain-theoretic rounded ideal completion.     

Note on Kocay's 3-Hypergraphs and Stockmeyer's Tournaments

In this note, we study the nonreconstructibility property through examples given by Stockmeyer (for tournaments) and Kocay (for 3-hypergraphs). Relating these examples we show how to obtain non (−1)-reconstructible ternary relations from particular non (−1)-reconstructible binary ones.     

Self-Complementary Non-Uniform Hypergraphs

Let V be a finite set. For a nonempty subset K of positive integers, a K-hypergraph on V is a hypergraph with vertex set V and edge set ${E=\bigcup_{k\in K}E_k}$ , where E _k is a nonempty set of k-subsets of V. We define the complement of a K-hypergraph (V, E) to be the K-hypergraph on V whose edge set consists of the subsets of V with cardinality in K which do not lie in E. A K-hypergraph is called self-complementary if it is isomorphic to its complement. The two extreme classes of self-complementary K-hypergraphs have been studied previously. When |K|?=?1 these are the self-complementary uniform hypergraphs, and when |K|?=?|V| ? 1, these are the so called ‘self-complementary hypergraphs’ studied by Zwonek. In this paper we determine necessary conditions on the order of self-complementary K-hypergraphs, and on the order of regular or vertex-transitive self-complementary K-hypergraphs, for various sets of positive integers K. We also present several constructions for K-hypergraphs to show that these necessary conditions are sufficient for certain sets K. In the language of design theory, the t-subset-regular self-complementary K-hypergraphs correspond to large sets of two isomorphic t-wise balanced designs, or t-partitions, in which the block sizes lie in the set K. Hence the results of this paper imply results in design theory.     

Connection Between Polynomial Optimization and Maximum Cliques of Non-Uniform Hypergraphs

In Motzkin and Straus (Canad. J. Math 498 17, 533540 1965) provided a connection between the order of a maximum clique in a graph G and the Lagrangian function of G. In Rota Bulò and Pelillo (Optim. Lett. 500 3, 287295 2009) extended the Motzkin-Straus result to r-uniform hypergraphs by establishing a one-to-one correspondence between local (global) minimizers of a family of homogeneous polynomial functions of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study similar optimization problems and obtain the connection to maximum cliques for {s, r}-hypergraphs and {p, s, r}-hypergraphs, which can be applied to obtain upper bounds on the Turán densities of the complete {s, r}-hypergraphs and {p, s, r}-hypergraphs.     

Torsion classes of abelian cyclically ordered groups

We introduce the notion of torsion class of abelian cyclically ordered groups; the definition is analogous to that used in the theory of lattice ordered groups. The collection T of all such classes is partially ordered by the class-theoretical inclusion. Though T is a proper class, we can apply the usual terminology for this partial order. We prove that T is a complete, infinitely distributive lattice having infinitely many atoms.     

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.     

The L* Partial Order on the Set of Group Matrices

In this paper,we use the L(o)wner partial order and the star partial order to introduce a new partial order (denoted by "L*") on the set of group matrices,and get some characteristics and properties of the new partial order.In particular,we prove that the L* partial order is a special kind of the core par-tial order and it is equivalent to the star partial order under some conditions.We also illustrate its difference from other partial orders with examples and find out under what conditions it is equivalent to other partial orders.     

The Ends of Algebras

We introduce the notion of ends for algebras. The definition is analogous to the one in geometric group theory. We establish some relations to growth conditions and cyclic cohomology.     

L—Fuzzy偏序集

借助L-fuzzy关系在L-fuzzy中集中引入L-fuzzy偏序,自然地有了L-fuzzy偏序集,进一步借助水平截集刻画了L-fuzzy偏序集。     

A sharp estimate for the Hilbert transform along finite order lacunary sets of directions

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler’s order is a central notion of the model theory of the 60s and 70s which compares first-order theories, and implicitly ultrafilters, according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the n-free k-hypergraphs studied by Hrushovski. This complexity reflects the difficulty of amalgamation and appears orthogonal to forking.     

Second order partial differential equations for gradients of orthogonal polynomials in two variables

In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one.     

The diamond partial order in rings

In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157–169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order.     

有限集合上偏序关系的矩阵判别方法

要从定义出发去判断集合X上的二元关系R是否是偏序关系比较困难,从关系矩阵出发,给出了偏序关系的矩阵判别方法,并对此方法进行了证明.     

Semi-tilting Modules and Mutation

We introduce the notion of semi-tilting modules and show that the class of basic semi-tilting modules is closed under mutation. Using this, we provide a partial answer to the Wakamatsu tilting conjecture. Also, we define a partial order on the set of equivalence classes of semi-tilting modules and show that this order is closely related to mutation for semi-tilting modules.