Obligatory subsystems of triple systems

We determine a class of triple systems such that each must occur in a triple system with uncountable chromatic number that omits $ \mathcal{T}_0 $ (the unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also show that consistently there are two finite triple systems such that they can separately be omitted by uncountably chromatic triple systems but not both.     

On subsystems of systems of sets

It is demonstrated that if {E_n} is a system of measurable sets satisfying the condition μE _n?μ₀>0, then it contains a subsystem \(\{ E_{n_k } \}\) independent in a certain generalized sense. A certain general assertion about the choice of subsystems is demonstrated in a preliminary way.     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

Perfect dexagon triple systems with given subsystems

The graph consisting of the six triples (or triangles) {a,b,c}, {c,d,e}, {e,f,a}, {x,a,y}, {x,c,z}, {x,e,w}, where a,b,c,d,e,f,x,y,z and w are distinct, is called a dexagon triple. In this case the six edges {a,c}, {c,e}, {e,a}, {x,a}, {x,c}, and {x,e} form a copy of K₄ and are called the inside edges of the dexagon triple. A dexagon triple system of order v is a pair (X,D), where D is a collection of edge disjoint dexagon triples which partitions the edge set of 3K_v. A dexagon triple system is said to be perfect if the inside copies of K₄ form a block design. In this note, we investigate the existence of a dexagon triple system with a subsystem. We show that the necessary conditions for the existence of a dexagon triple system of order v with a sub-dexagon triple system of order u are also sufficient.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Steiner triple systems with disjoint or intersecting subsystems

The existence of incomplete Steiner triple systems of order υ having holes of orders w and u meeting in z elements is examined, with emphasis on the disjoint (z = 0) and intersecting (z = 1) cases. When and , the elementary necessary conditions are shown to be sufficient for all values of z. Then for and υ “near” the minimum of , the conditions are again shown to be sufficient. Consequences for larger orders are also discussed, in particular the proof that when one hole is at least three times as large as the other, the conditions are again sufficient. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 58–77, 2000     

Analysis of channel and disk subsystems in computer systems

The information processing industry has shifted from the conventional mathematical computation to information management. Interactive use of a system with large data bases is now widespread, and the data base has become one of the key factors central to an overall system design. This, in turn, has prompted the development of new peripheral devices.The paper focuses on developing and analyzing stochastic models for disk storage systems. A general model is developed to describe the sequence of operations required for accessing a record on a disk. The model is then used to analyze how technological and architectural changes affect system performance. As concrete examples, performance comparison of IBM 3330 disk system with IBM 3380 disk system will be conducted. All analytical results are validated via simulation.Supported by the National Science Foundation under Grant No. ECS-8404071.     

On the existence of nearly Kirkman triple systems with subsystems

It is proved in this paper that the necessary and sufficient conditions for the existence of an incomplete nearly Kirkman triple system INKTS(u, v) are u ≡ v ≡ 0 (mod 6), u ≥ 3v. As a consequence, we obtain a complete solution to the embedding problem for nearly Kirkman triple systems.      

Further results on nearly Kirkman triple systems with subsystems

In this paper we further discuss the embedding problem for nearly Kirkman triple systems and get the result that: (1) For u≡v≡0 (mod 6), v78, and u3.5v, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v=18,24,30,36,42,48,54,60,66 or 72, there exists an NKTS(u) containing a sub-NKTS(v) if and only if u≡0 (mod 6) and u3v.     

Stability of switched nonlinear time-delay systems with stable and unstable subsystems

This paper investigates the stability of switched nonlinear time-delay systems with stable and unstable subsystems. Several stability criteria are presented by resorting to novel inequality technique and average dwell time approach, which relax the assumptions that all subsystem matrices are commutative pairwise and Hurwitz stable. Finally, two numerical examples are provided to illustrate the effectiveness of the theoretical results.     

A new approach to the construction of subsystems of complex root systems

Dynkin has shown how subsystems of real root systems may be constructed. As the concept of subsystems of complex root systems is not as well developed as in the real case, in this paper we give an algorithm to classify the proper subsystems of complex proper root systems. Furthermore, as an application of this algorithm, we determine the proper subsystems of imprimitive complex proper root systems. These proper subsystems are useful in giving combinatorial constructions of irreducible representations of properly generated finite complex reflection groups.     

Steiner triple systems of order 19 and 21 with subsystems of order 7

Steiner triple systems (STSs) with subsystems of order 7 are classified. For order 19, this classification is complete, but for order 21 it is restricted to Wilson-type systems, which contain three subsystems of order 7 on disjoint point sets. The classified STSs of order 21 are tested for resolvability; none of them is doubly resolvable.     

Tracking control for switched time-varying delays systems with stabilizable and unstabilizable subsystems

We investigate the tracking control problem for switched linear time-varying delays systems with stabilizable and unstabilizable subsystems. Sufficient conditions for the solvability of the tracking control problem are developed. The tracking control problem of a switched time-varying delays system with stabilizable and unstabilizable subsystems is solvable if the stabilizable and unstabilizable subsystems satisfy certain conditions and admissible switching law among them. Average dwell time approach and piecewise Lyapunov functional methods are utilized to the stability analysis and controller design. By introducing the integral controllers and free weighting matrix scheme, some restricted assumptions imposing on the switched systems are avoided. A simulation example shows the effectiveness of the proposed method.