This presentation is an exposition of an application of the theory of recurrence relations to enumerating strings over an alphabet with a forbidden factor (consecutive substring). As an illustration we examine the case of binary strings with a forbidden factor of k consecutive symbols 1 for given k, using generating function techniques that deserve to be better known.
For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG+, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.
We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.
Bonus-Malus systems are widely used in the insurance business. For the operation of such systems transition rules and premium scales need to established. Optimization of these systems usually means the calculation of appropriate premium scales while treating transition rules external parameters. In this paper, on the contrary, we show how optimal transition rules can be determined for a given set of premium scales. To this end, integer programming (IP) is used as a basic tool. Numerical examples are also presented to demonstrate the viability of our approach.
In this paper, we modify the iteration process of Agarwal et al. (J. Nonlinear Convex Anal. 8(1) (2007), 61–79) to three multi-valued mappings and prove the strong and △-convergence theorems of this iteration in a hyperbolic space. Our theorems extend and improve some recent results announced in the current literature.
