When does a notch behave like a crack?

The characteristic asymptotic fields at the tip of sharp, semi-infinite cracks and notches are first compared with corresponding features present in selected finite bodies (edge cracks and notches). This gives an explicit view of the gradual divergence of the semi-infinite and finite problem solutions as the observation point becomes remote from the tip. Hence, upper bounds for the local plastic zone to be characterized by the singular field are known. Asymptotic solutions for semi-infinite rounded features are introduced, whose remote fields may be matched to the sharp singular fields through the medium of the corresponding generalized stress intensity factor. Thus the semi-infinite sharp and rounded problems converge remotely but diverge as the apex of the feature is approached. This comparison sets a lower bound for loads at which the outer boundary of the plastic zone is characterized by the singular field. Thus, the range of loads for the plastic zones to be controlled by the singular solutions are derived. We then proceed to compare critically the nature of the semi-infinite sharp notch and semi-infinite crack states of stress, defining the circumstances in which these are alike. All these elements considered together enable the closeness of various notch plastic zones to that of the classical semi-infinite crack to be gauged.

On a distributed clustering process of Coffman, Courtois, Gilbert and Piret

Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution.We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times?Coffmanet al.concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known.We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from afinitecomputation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.

On a distributed clustering process of Coffman, Courtois, Gilbert and Piret

Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution. We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times? Coffman et al. concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known. We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from a finite computation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.

On the full information best-choice problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

On the full information best-choice problem

We introduce the optimal stopping problem of an infinite sequence of records associated with a planar Poisson process. This problem serves as a limiting form of the classical full information best-choice problem. A link between the finite problem and its limiting form is established via embedding n i.i.d. observations into the planar process.

Generalizing the secretary problem

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.

Generalizing the secretary problem

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.

Cultural Value of Mathematics

“The intellect never slumbers,” but is ever searching for knowledge and truth. It is ever groping about in the darkness of error and of doubt, and if truly honest in its search, uses every light which an all wise and loving Master has given it to detect the slightest flaw in every finite problem which is presented to it. Its restlessness continues until the goal of absolute certitude is reached.