The hyperplanes of the symplectic dual polar space DW (5, q) arising from embedding, the so-called classical hyperplanes of DW (5, q), have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of DW (5, q). If q is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of DW (5, q). If q is odd, then w...

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envyfree, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by ...

Fair Division, unlike most research on fairness in the social sciences and mathematics, is devoted solely to the analysis of constructive procedures for actually dividing things up and resolving disputes, including...

We propose a theoretical framework for odor classification in the olfactory system of insects. The classification task is accomplished in two steps. The first is a transformation from the antennal lobe to the intrinsic Kenyon cells in the mushroom body. This transformation into a higher-dimensional space is an injective function and can be implemented without any type of learning at the synapti...

A (k, n) secret image sharing method shares a protected image in n shared images and the protected image can be reconstructed by k shared images. This study solves the secret image sharing problem by a geometry secret sharing strategy named the Blakley scheme. The protected image is first partitioned into non-overlapping sets of k pixels. Each set of k pixels forms a point under a k-dimensional...

In this paper we investigate the combinatorial complexity of an algorithm to determine the geometry and the topology related to an arrangement of hyperplanes in multi-dimensional Euclidean space from the “probing” on the arrangement. The “probing” by a flat means the operation from which we can obtain the intersection of the flat and the arrangement. For a finite set H of hyperplanes in Ed, we ...