Most of the data on the relat ive efficiency of dif ferent implementations of the alpha-beta a l gorithm is neither readily available nor in a form suitable for easy comparisons. In the present study four enhancements to the alpha-beta algorithm—iterative deepening, aspiration search, memory tables and principal variation search—are compared separately and in various combinations to determine t...

Most practitioners use a variant of the Alpha-Beta algorithm, a simple depth-first procedure, for searching minimax trees. SSS*, with its best-first search strategy, reportedly offers the potential for more efficient search. However, the complex formulation of the algorithm and its alleged excessive memory requirements preclude its use in practice. For two decades, the search efficiency of “sma...

This paper presents a numerical method for finding the solution of Plateau’s problem in parametric form. Using the properties of minimal surfaces we succeded in transfering the problem of finding the minimal surface to a problem of minimizing a functional over a class of scalar functions. A numerical method of minimizing a functional using the first variation is presented and convergence is pro...

Consider a closed lipid membrane (vesicle), modeled as a two-dimensional surface, described by a geometrical hamiltonian that depends on its extrinsic curvature. The vanishing of its first variation determines the equilibrium configurations for the system. In this paper, we examine the second variation of the hamiltonian about any given equilibrium, using an explicitly surface covariant geometr...

A sequence x1, . . . , xn, . . . of discrete-valued observations is generated according to some unknown probabilistic law (measure) μ. After observing each outcome, it is required to give the conditional probabilities of the next observation. The realizable case is when the measure μ belongs to an arbitrary but known class C of process measures. The nonrealizable case is when μ is completely ar...

The Denjoy-Young-Saks Theorem from classical analysis states that for an arbitrary function f : R → R, the Denjoy alternative holds outside a null set. This means that for almost every real x, either the derivative of f exists at x, or the derivative fails to exist in the worst possible way: the limit superior of the slopes around x equals +∞, and the limit inferior −∞. Algorithmic randomness a...

We consider the problem of scheduling trains and containers (or trucks and pallets) between a depot and a destination. Goods arrive at the depot dynamically over time and have distinct due dates at the destination. There is a fixed-charge transportation cost for each vehicle, and each vehicle has the same capacity. The cost of holding goods may differ between the depot and the destination. The ...

In lecture 8 we have formulated the condition of static equilibrium of bodies and structures by studying a small change (variation) of the total potential energy. The system was said to be in equilibrium if the first variation of the total potential energy vanishes. The analysis did not say anything about the stability of equilibrium. The present lecture will give an answer to that question by ...

In this note, we establish the first variation formula of the adjusted log entropy functional Ya introduced by Ye in [14]. As a direct consequence, we also obtain the monotonicity of Ya along the Ricci flow. Various entropy functionals play crucial role in the singularity analysis of Ricci flow. Let (M, g(t)) be a smooth family of Riemannian metrics on a closed manifold M and suppose g(t) is a ...

We prove that an m-dimensional minimal variety in a Riemannian manifold cannot touch the boundary at a point where the sum of the smallest m principal curvatures is greater than 0. We also prove an analogous result for varieties with bounded mean curvature. Let N be a smooth Riemannian manifold with boundary. In general, N need not be complete. Suppose X is a compactly supported C tangent vecto...