Self-induced Separation. III

The theory of Stewartson & Williams for the self-inducedseparation of a laminar boundary layer on a flat plate in asupersonic mainstream is extended to include a further termin the expansion for the pressure. The relevant equations aresolved numerically and a comparison made with available experimentalresults.     

Historical systems as complex systems

All social systems are simultaneously historical and systemic. They have rules that govern their operation which are reflected in cyclical rhythms; they have irreversible patterns of development which are reflected in their secular trends, and which account for their eventual demise as systems. There are two varieties of complex historical systems: world-empires and world-economies. The latter variety has come to dominate in the period since 1500, leading to the elimination of all other varieties and creating the new situation of a planet with only one existing historical system. The consequences are explored.     

Gradient systems and gradient-like systems

We briefly discuss existing results on convergence and nonconvergence to equilibrium of bounded solutions of gradient systems and gradient-like systems. We especially raise the question of the influence of the ambient metric. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On set systems not containing delta systems

Some constructions of systems ofn-sets not containing (k) systems are given. The constructions were produced by both analytical and computational methods.Research supported by Office of Naval Research Grant No. N0014-88-K0211.     

Dual systems of algebraic iterated function systems

We study a class of graph-directed iterated function systems on R$\mathbb{R}$ with algebraic parameters, which we call algebraic GIFS. We construct a dual IFS of an algebraic GIFS, and study the relations between the two systems. We determine when a dual system satisfies the open set condition, which is fundamental. For feasible Pisot systems, we construct the left and right Rauzy–Thurston tilings, and study their multiplicities and decompositions. We also investigate their relation with codings space, domain-exchange transformation, and the Pisot spectrum conjecture. The dual IFS provides a unified and simple framework for Rauzy fractals, β-tilings and related studies, and allows us gain better understanding.     

Expert systems for planning and scheduling manufacturing systems

In this paper, research and applications of expert systems in production planning and scheduling are reviewed. Components of expert systems are briefly discussed. Relationship between expert system and operations research approaches are presented. Integration of operations research and expert system techniques is explored.     

Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Embedding 5-cycle systems into pentagon triple systems

We show that the spectrum for pentagon triple systems is the set of all n≡1,15,21 or . We then construct a 5-cycle system of order 10n+1 which can be embedded in a pentagon triple system of order 30n+1 and also construct a 5-cycle system of order 10n+5 which can be embedded in a pentagon triple system of order 30n+15, with the possible exception of embedding a 5-cycle system of order 21 in a pentagon triple system of order 61.     

Duality theorems for linear systems and convex systems

We show that, if (F →^uX) is a linear system, $\text{Ω ? X}$ a convex target set and $\text{h: X →}\text{R}\text{?}$ a convex functional, then, under suitable assumptions, the computation of inf $\text{h({y ? F ¦ u(y) ? Ω}}$) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of $\text{u}{}^1\text{(X}{}^1\text{)}$, or of the infima on F of Lagrangians, involving elements of $\text{u}{}^1\text{(X}{}^1\text{)}$. Also, we prove similar results for a convex system (F →^uX) and the convex cone Ω of all non-positive elements in X.     

Disjoint systems

A disjoint system of type (?, ?, k, n) is a collection ?? = {??₁,…, ??_m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ??_i and ??_j of distinct members of ?? and for every A ? ??_i there exists a B ? ??_j that does not intersect A. Let D_n (?, ?, k) denote the maximum possible cardinality of a disjoint system of type (?, ?, k, n). It is shown that for every fixed k ? 2,. This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.     

Orthogonal systems

An equidistant permutation array (E.P.A.)A(r, v) is av × r array in which every row is a permutation of the integers 1, 2, ,r such that any two distinct rows have precisely columns in common. In this paper we introduce the concept of orthogonality for E.P.A.s. A special case of this is the well known idea of a set of pairwise orthogonal latin squares. We show that a set of these arrays is equivalent to a particular type of resolvable (r, )-design. It is also shown that the cardinality of such a set is bounded byr – with the upper bound being obtained only if = 0. A brief survey of related orthogonal systems is included. In particular, sets of pairwise orthogonal symmetric latin squares, sets of orthogonal Steiner systems and sets of orthogonal skeins.     

Toeplitz-Hausdorff systems

The numerical range of an n × n matrix T is the image of T under a certain set of linear functionals—a set that comprises the extreme points among the states (i.e., norm-one, positive linear functionals) on the n × n matrices—and is convex, by the Toeplitz-Hausdorff theorem. One can view this convexity as a consequence of T's numerical range being equal to a manifestly convex set, the image of T under all states. Taking this view leads us to ask whether a similar result holds when we replace the n × n matrices by a finite dimensional Banach space

, the states by a closed, convex subset Σ of

1, and the extreme states by the extreme points of Σ. When it does, we call the pair (

, Σ) a Toeplitz-Hausdorff system. In this paper, we show that if

is what we term a nullifying subspace of the n×n matrices, and if Σ is the closed unit ball in

1, then (

, Σ) is a Toeplitz-Hausdorff system. (Both the upper and lower triangular matrices form nullifying subspaces.)     

Canonical systems

In this paper, we study canonical systems and pose the problem of constructing a quasihomogeneous canonical system, i.e., the canonical system whose spectral problem has the same solution as that of the spectral problem of some homogeneous canonical system. We present a rational algorithm for constructing a quasihomogeneous canonical systems.