During the 20th century there has been an incredible progress in solving theoretically hard problems in practice. One of the most prominent examples is the DPLL algorithm and its derivatives to solve the Boolean satisfiability problem, which can handle instances with millions of variables and clauses in reasonable time, notwithstanding the theoretical difficulty of solving the problem. Despite ...

In this note we present a new characterization of Bernstein operators by showing that they are the only solution of a certain extremal relation. © 2006 Elsevier Inc. All rights reserved.

In this paper we examine periodic integrodifferential equations in Banach spaces. When the cone is regular, we prove two existence theorems for the extremal solutions in the order interval determined by an upper and a lower solution. Both theorems use only the order structure of the problem and no compactness condition is assumed. In the last section we ask the cone to be only normal but we imp...

We study the existence and computation of extremal solutions of a system of inequations de ned over lattices. Using the Knaster-Tarski xed point theorem, we obtain su cient conditions for the existence of supremal as well as in mal solution of a given system of inequations. Iterative techniques are presented for the computation of the extremal solutions whenever they exist, and conditions under...

Engineering optimization problems usually have several conflicting objectives, such that no single solution can be considered optimum with respect to all objectives. In recent years, many efforts have focused on hybrid metaheuristic approaches for their robustness and efficiency to solve the above-mentioned multiobjective optimization problems (MOPs). This paper proposes a novel hybrid algorith...

This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach used to establish the results for degree 6 and 7 has been extended successfully to degree 8 and 9. Candidate graphs are defined as functions of the diameter fo...

The classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f ′′+A(z)f ′+ B(z)f = 0 are of infinite order is discussed. Two distinct approaches are used. In the first approach the coefficient A(z) itself is a solution of a differential equation w′′ + P (z)w = 0, where P (z) is a polynomial. This assumption yields stability o...

We discuss the Siciak-Zaharjuta extremal function of a real convex body in Cn, a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic ...

Two old conjectures from problem sections, one of which from SIAM Review, concern the question of finding distributions that maximize P(Sn ≤ t), where Sn is the sum of i.i.d. random variables X1, . . . , Xn on the interval [0, 1], satisfying E [X1] = m. In this paper a Lagrange multiplier technique is applied to this problem, yielding necessary conditions for distributions to be extremal, for a...

We discuss the Siciak-Zaharjuta extremal function of a real convex body in C n , a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphi...