Generating all subsets of a finite set with disjoint unions

If X is an n-element set, we call a family G⊂PX a k-generator for X if every x⊂X can be expressed as a union of at most k disjoint sets in G. Frein, Lévêque and Seb? conjectured that for n>2k, the smallest k-generators for X are obtained by taking a partition of X into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We prove this conjecture for all sufficiently large n when k=2, and for n a sufficiently large multiple of k when k?3.     

Structure and stability of triangle-free set systems

We define the notion of stability for a monotone property of set systems. This phenomenon encompasses some classical results in combinatorics, foremost among them the Erdos-Simonovits stability theorem. A triangle is a family of three sets such that , , are each nonempty, and . We prove the following new theorem about the stability of triangle-free set systems.

Fix . For every , there exist and such that the following holds for all : if and is a triangle-free family of -sets of containing at least members, then there exists an -set which contains fewer than members of .

This is one of the first stability theorems for a nontrivial problem in extremal set theory. Indeed, the corresponding extremal result, that for every triangle-free family of -sets of has size at most , was a longstanding conjecture of Erdos (open since 1971) that was only recently settled by Mubayi and Verstraëte (2005) for all .

     

The Unbalance of Set Systems

The unbalance of an intersecting family is defined as , where is the maximum degree of i.e. the maximum of over all vertices x. We show that the unbalance of a k-uniform intersecting family is at most when n ≥ 6k ³ and we determine all families achieving this bound.     

Further results on large set of disjoint pure Mendelsohn triple systems

In this article, we study a large set of disjoint pure Mendelsohn triple systems “with holes” (briefly LPHMTS), which is a generalization of large set of disjoint pure Mendelsohn triple systems (briefly LPMTS), and give some recursive constructions on LPHMTS. Using these constructions, we show that there exists LPMTS(2ⁿ + 2) for any n ≠ 2. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 274–290, 2000     

On set systems with a threshold property

For n,k and t such that 1<t<k<n, a set F of subsets of [n] has the (k,t)-threshold property if every k-subset of [n] contains at least t sets from F and every (k-1)-subset of [n] contains less than t sets from F. The minimal number of sets in a set system with this property is denoted by m(n,k,t). In this paper we determine m(n,4,3)exactly for n sufficiently large, and we show that m(n,k,2) is asymptotically equal to the generalized Turán number T_k-1(n,k,2).     

On large sets of disjoint Kirkman triple systems

In this paper, we introduce LR(u) designs and use these designs together with large sets of Kirkman triple systems (LKTS) and transitive KTS (TKTS) of order v to construct an LKTS(uv). Our main result is that there exists an LKTS(v) for v∈{3nm(2·13k+1)t;n?1,k?1,t=0,1,m∈{1,5,11,17,25,35,43}}.     

Fractional variational integrators for fractional variational problems

In this paper, the fractional variational integrators for a class of fractional variational problems are developed. The fractional discrete Euler-Lagrange equation is obtained. Based on the Grünwald-Letnikov method and Diethelm’s fractional backward differences, some fractional variational integrators are presented and the fractional variational errors are discussed. Some numerical examples are presented to illustrate these results.     

Maximum graphs with a unique minimum dominating set

We present a conjecture on the maximum number of edges of a graph that has a unique minimum dominating set. We verify our conjecture for some special cases and prove a weakened version of this conjecture in general.     

Incongruent restricted disjoint covering systems

We define an incongruent restricted disjoint covering system on [1,n] as a set of congruence classes such that each integer in the interval [1,n] belongs to exactly one class, and each class contains at least two members of the interval. In this paper we report some computational and structural results and present some open problems concerning such systems.     

Finding dominators via disjoint set union

The problem of finding dominators in a directed graph has many important applications, notably in global optimization of computer code. Although linear and near-linear-time algorithms exist, they use sophisticated data structures. We develop an algorithm for finding dominators that uses only a “static tree” disjoint set data structure in addition to simple lists and maps. The algorithm runs in near-linear or linear time, depending on the implementation of the disjoint set data structure. We give several versions of the algorithm, including one that computes loop nesting information (needed in many kinds of global code optimization) and that can be made self-certifying, so that the correctness of the computed dominators is very easy to verify.     

Updating attribute reduction in incomplete decision systems with the variation of attribute set

In rough set theory, attribute reduction is a challenging problem in the applications in which data with numbers of attributes available. Moreover, due to dynamic characteristics of data collection in decision systems, attribute reduction will change dynamically as attribute set in decision systems varies over time. How to carry out updating attribute reduction by utilizing previous information is an important task that can help to improve the efficiency of knowledge discovery. In view of that attribute reduction algorithms in incomplete decision systems with the variation of attribute set have not yet been discussed so far. This paper focuses on positive region-based attribute reduction algorithm to solve the attribute reduction problem efficiently in the incomplete decision systems with dynamically varying attribute set. We first introduce an incremental manner to calculate the new positive region and tolerance classes. Consequently, based on the calculated positive region and tolerance classes, the corresponding attribute reduction algorithms on how to compute new attribute reduct are put forward respectively when an attribute set is added into and deleted from the incomplete decision systems. Finally, numerical experiments conducted on different data sets from UCI validate the effectiveness and efficiency of the proposed algorithms in incomplete decision systems with the variation of attribute set.     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

Nonlinear fractional cone systems with the Caputo derivative

Nonlinear fractional cone systems involving the Caputo fractional derivative are considered. We establish sufficient conditions for the existence of at least one cone solution to such systems. Sufficient conditions for the unique existence of the cone solution to a nonlinear fractional cone system are given.     

Synchronization of nonlinear fractional order systems

This paper deals with the master-slave synchronization scheme for partially known nonlinear fractional order systems, where the unknown dynamics is considered as the master system and we propose the slave system structure which estimates the unknown state variables. For solving this problem we introduce a Fractional Algebraic Observability (FAO) property which is used as a main tool in the design of the master system. As numerical examples we consider a fractional order Rössler hyperchaotic system and a fractional order Lorenz chaotic system and by means of some simulations we show the effectiveness of the suggested approach.     

Fractional dynamics of populations

Nature often presents complex dynamics, which cannot be explained by means of ordinary models. In this paper, we establish an approach to certain fractional dynamic systems using only deterministic arguments. The behavior of the trajectories of fractional non-linear autonomous systems around the corresponding critical points in the phase space is studied. In this work we arrive to several interesting conclusions; for example, we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to (or away from) the critical point. Such property could contribute to faithfully represent the anomalous reality of the competition among some species (in cellular populations as Cancer or HIV). We use classical models, which describe dynamics of certain populations in competition, to give a justification of the possible interest of the corresponding fractional models in biological areas of research.