Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals
of integer lengths, remains open. More generally an integral point set $mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $mathbb{F}_q^2$ over a finite field $mathbb{F}_q$, where the fo
rmally defined squared Euclidean distance of every pair of points is a square in $mathbb{F}_q$. It turns out that integral point sets over $mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $mathbb{F}_q$ for $qle 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.
We propose the new Top-Dog-Index to quantify the historic deviation of the supply data of many small branches for a commodity group from sales data. On the one hand, the common parametric assumptions on the customer demand distribution in the literat
ure could not at all be supported in our real-world data set. On the other hand, a reasonably-looking non-parametric approach to estimate the demand distribution for the different branches directly from the sales distribution could only provide us with statistically weak and unreliable estimates for the future demand. Based on real-world sales data from our industry partner we provide evidence that our Top-Dog-Index is statistically robust. Using the Top-Dog-Index, we propose a heuristics to improve the branch-dependent proportion between supply and demand. Our approach cannot estimate the branch-dependent demand directly. It can, however, classify the branches into a given number of clusters according to an historic oversupply or undersupply. This classification of branches can iteratively be used to adapt the branch distribution of supply and demand in the future.
We propose the new Top-Dog-Index, a measure for the branch-dependent historic deviation of the supply data of apparel sizes from the sales data of a fashion discounter. A common approach is to estimate demand for sizes directly from the sales data. T
his approach may yield information for the demand for sizes if aggregated over all branches and products. However, as we will show in a real-world business case, this direct approach is in general not capable to provide information about each branchs individual demand for sizes: the supply per branch is so small that either the number of sales is statistically too small for a good estimate (early measurement) or there will be too much unsatisfied demand neglected in the sales data (late measurement). Moreover, in our real-world data we could not verify any of the demand distribution assumptions suggested in the literature. Our approach cannot estimate the demand for sizes directly. It can, however, individually measure for each branch the scarcest and the amplest sizes, aggregated over all products. This measurement can iteratively be used to adapt the size distributions in the pre-pack orders for the future. A real-world blind study shows the potential of this distribution free heuristic optimization approach: The gross yield measured in percent of gross value was almost one percentage point higher in the test-group branches than in the control-group branches.
