Minimizing the number of lattice points in a translated polygon

The parametric lattice-point counting problem is as follows: Given an integer matrix $A \in Z^{m \times n}$, compute an explicit formula parameterized by $b \in R^m$ that determines the number of integer points in the polyhedron $\{x \in R^n : Ax \leq b\}$. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok's algorithm have been shown to solve this problem in polynomial time if the number $n$ of columns of $A$ is fixed. Central to our investigation is the following question: Can one also efficiently determine a parameter $b$ such that the number of integer points in $\{x \in R^n : Ax \leq b\}$ is minimized? Here, the parameter $b$ can be chosen from a given polyhedron $Q \subseteq R^m$. Our main result is a proof that finding such a minimizing parameter is $NP$-hard, even in dimension 2 and even if the parametrization reflects a translation of a 2-dimensional convex polygon. This result is established via a relationship of this problem to arithmetic progressions and simultaneous Diophantine approximation. On the positive side we show that in dimension 2 there exists a polynomial time algorithm for each fixed $k$ that either determines a minimizing translation or asserts that any translation contains at most $1 + 1/k$ times the minimal number of lattice points

Some aspects of estate duty on deceased estates in the Republic of South Africa: with special reference to the problems and effects of double taxation occuring in relation to the German Erbschaftsteuergesetz (Inheritance Tax Act)

'In this world nothing can be said to be certain, except death and taxes'. If one agreed with this statement, made by Benjamin Franklin one year prior to his death, one could come to the ultimate conclusion that there has to be a special certainty about capital transfer taxes, estate duties, inheritance taxes or other death duties levied on deceased estates. There seems to be no reason whatsoever to doubt the final truth of the first part of Franklin's statement. But is it also right to assume that there is a certainty about taxes or, what is more, to draw the conclusion that there could be a special certainty about the different types of taxes connected with the death of an individual? On the one hand it seems to be true. There are only a few countries throughout the world levying very limited or no taxes at all, and if one forgets about a few tax havens, like the Caymen Islands for example, one is also inclined to agree with the latter part of Franklin's statement - there seems to be certainty about the levying of taxes. But, on the other hand, is the amount of taxes, especially the amount of capital transfer taxes, estate duties, inheritance taxes or other death duties levied on a persons death, always that certain? The avoidance of double taxation has been an objective of most of the world's nations since the negotiation of the first Income Tax Convention in the middle of the nineteenth century. The development of most of the world's countries into modern industrial nations, with trade links all over the world, gradually demonstrated the problems which can arise if one country levies taxes without considering the levying of an equal or similar tax in another country. Today the negative consequences of double taxation on the movement of goods, services and capital are widely understood by modern states enacting tax systems for their citizens

2 kirja Karl Morgensternile, Berlin

http://tartu.ester.ee/record=b1752663~S6*es

On sub-determinants and the diameter of polyhedra

We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we show that the diameter of P is bounded by O(\Delta^2 n^4 log n\Delta). If P is bounded, then we show that the diameter of P is at most O(\Delta^2 n^3.5 log n\Delta). For the special case in which A is a totally unimodular matrix, the bounds are O(n^4 log n) and O(n^3.5 log n) respectively. This improves over the previous best bound of O(m^16 n^3 (log mn)^3) due to Dyer and Frieze

Diameter of Polyhedra: Limits of Abstraction

We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear lower bound

Ueber die Mischung der feuerbeständigen Alkalien : eine Inaugural-Dissertation

http://tartu.ester.ee/record=b1900718~S1*es