The maximum amplitude cost functional in linear systems

The maximum amplitude cost of a control function u(t) taken to be ess sup g(t, u(t)), where g(t, u) is a given function. (A particular example is g(t, u) = the norm of u.) We consider linear systems with this cost functional. The existence of optimal control is proved, and it is shown that the ess sup is uniformly essential with respect to the optimal controls. Properties of the extended attainable set are discussed and compared with the case of an integral cost. Finally, we show in what sense a cost functional of the form $\text{(∝ g(t, u(t))}{}^{\mathrm{q}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>q</mtext>}}$ approximates the ess sup cost functional.     

Ground-state energies of atomic chains in superstrong magnetic fields

The ground-state energies of hydrogenic and helium chains in superstrong magnetic fields (B ≈? 10⁷?10⁹ Tesla) have been calculated within the single-particle scheme of a heuristic density functional method. Applying the local density approximation (LDA ), the simple Dirac exchange functional was used neglecting correlation effects. The equations were solved iteratively using a basis set of Landau functions in the transverse directions and directly integrating along the longitudinal direction. We obtained binding lengths and, by comparing the binding energies of the chains with the ground-state energies of single atoms, we found condensation energies. © 1995 John Wiley & Sons, Inc.     

An EOQ Model with Substitutions Between Products

In this paper, we present an economic order quantity (EOQ) model when two products are required, and one can be substituted for the other, if necessary, at a given unit cost. We consider three cases: (i) when there is no substitution between the products, (ii) when there is full substitution between the products, and (iii) when there is partial substitution between the products. In a deterministic setting with proportional substitution costs, we would expect to find full substitution or no substitution being optimal, depending on the cost parameters. However, we observe that full substitution is never optimal; only partial substitution or no substitution may be optimal. This result can best be explained due to the non-linearity of the decision variables in the total cost expression. Finally, we present an algorithm to compute the optimal order quantities.     

On the existence of special depth first search trees

The Depth First Search (DFS) algorithm is one of the basic techniques that is used in a very large variety of graph algorithms. Most applications of the DFS involve the construction of a depth-first spanning tree (DFS tree). In this paper, we give a complete characterization of all the graphs in which every spanning tree is a DFS tree. These graphs are called Total-DFS-Graphs. We prove that Total-DFS-Graphs are closed under minors. It follows by the work of Robertson and Seymour on graph minors that there is a finite set of forbidden minors of these graphs and that there is a polynomial algorithm for their recognition. We also provide explicit characterizations of these graphs in terms of forbidden minors and forbidden topological minors. The complete characterization implies explicit linear algorithm for their recognition. In some problems the degree of some vertices in the DFS tree obtained in a certain run are crucial and therefore we also consider the following problem: Let G = (V,E) be a connected undirected graph where |V| = n and let d ? Nⁿ be a degree sequence upper bound vector. Is there any DFS tree T with degree sequence d _T that violates d (i.e., d _T ≤ d , which means: E j such that d _T(j) > d (j))? We show that this problem is NP-complete even for the case where we restrict the degree of only on specific vertex to be less than or equal to k for a fixed k ≥ 2 (i.e., d = (n - 1, ?, n - 1, k, n - 1, ?, n - 1)). 0 1995 John Wiley & Sons, Inc.     

Asymptotic behavior of the Weber location problem on the plane

The asymptotic behavior of the Weber location problem is investigated. We consider problems wheren demand points are randomly generated in a unit disk by a uniform distribution and all weights are equal to one. The main result of the paper is that the probability that the optimal solution be on a demand point is approximately 1/n. Additional results for a largen: the optimal solution converges almost surely to the center of the disk; the difference between the optimal value of the objective function and the minimal value of the objective function on a demand point converges to 1/2.     

Convergence to convex compact sets in infinite dimensions

We provide a simple test for the convergence of a sequence of subsets in an infinite-dimensional space to a compact convex one. The test reduces, roughly, the convergence problem to finite dimensions. Applications are portrayed demonstrating how convexification phenomena established in finite dimensions are easily generalized to infinite dimensions.     

Relaxed Multifunctions and Young Multimeasures

Young-measure type limits of multifunctions are examined. The relaxed limit via selections is a multimeasure, while the relaxation as a set-valued point function yields a Young measure on sets. The paper verifies that the two processes are equivalent in the sense that the limit multimeasure consists of all the distributions selectionable with respect to the Young measure on sets. The main tool is the characterization of selectionable distributions. An illustrative application of the equivalence result is provided.