Committee Size and Smart Growth: An Optimal Solution

Wisconsin is one of many states that have enacted a “Smart Growth Initiative” law that requires inclusion of the public in the creation and development of a Comprehensive Plan. One implication of public participation is the strategic development of a comprehensive planning committee. Two crucial decisions occur when the committee is formed: the size of the committee and the composition of the committee. This paper models a relation between committee size and the accuracy of plan, as well as the relationship between the inclusion of experts, whether paid consultants or planners, and the quality of the outcome. Based on a survey of committee members, we test the relationship between the participants’ observations of quality and group size and composition, analyzing the tradeoff between the size of the group with the perception of quality of the decision

Separating invariants for the basic G_a-actions

We explicitly construct a finite set of separating invariants for the basic \Ga-actions. These are the finite dimensional indecomposable rational linear representations of the additive group \Ga of a field of characteristic zero, and their invariants are the kernel of the Weitzenb\"ock derivation $D_{n}=x_{0}\frac{\partial}{\partial{x_{1}}}+...+ x_{n-1}\frac{\partial}{\partial{x_{n}}}$.Comment: 10 page

On Cohen-Macaulayness and depth of ideals in invariant rings

We investigate the presence of Cohen-Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a $p$-group is not Cohen-Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen-Macaulay factorial rings cannot contain Cohen-Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen-Macaulay, extending a result of Fleischmann.Comment: 9 page

The separating variety for the basic representations of the additive group

For a group $G$ acting on an affine variety $X$, the separating variety is the closed subvariety of $X\times X$ encoding which points of $X$ are separated by invariants. We concentrate on the indecomposable rational linear representations $V_n$ of dimension $n+1$ of the additive group of a field of characteristic zero, and decompose the separating variety into the union of irreducible components. We show that if $n$ is odd, divisible by four, or equal to two, the closure of the graph of the action, which has dimension $n+2$, is the only component of the separating variety. In the remaining cases, there is a second irreducible component of dimension $n+1$. We conclude that in these cases, there are no polynomial separating algebras.Comment: 14 page

Degree bounds for separating invariants

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v'). It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree <= |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) <= b(G) <= [G:H] b(H)$, and that b(G) <= b(G/H) b(H)$ in case H is normal. Moreover, we calculate b(G) for some specific finite groups.Comment: 11 page

Reproductive behaviour of migrant women in Germany: Data, patterns and determinants

This paper examines the fertility of female migrants in Germany. After introducing major hypotheses on migrant fertility we give an overview on German datasets that are available for migrant fertility research. Finally, descriptive and multivariate analyses based on the "Sample Survey of Selected Migrant Groups in Germany (RAM)" are presented. Migrant fertility in Germany differs according to the country of origin: among major migrant groups analysed, Turkish women show the highest and Polish women the lowest fertility level. Multivariate analysis shows that the existence of children born in the country of origin has a strong increasing effect on migrant fertility. Besides, migrant women with German partners have a lower fertility than women with non-German partners. Furthermore, the fertility of Muslim women is elevated when compared with other religious groups. In contrast, emotional ties with the country of origin and the level of native and German language skills show no influence on migrants' fertility.

Invariants of the dihedral group D2pD_{2p} in characteristic two

We consider finite dimensional representations of the dihedral group $D_{2p}$ over an algebraically closed field of characteristic two where $p$ is an odd integer and study the degrees of generating and separating polynomials in the corresponding ring of invariants. We give an upper bound for the degrees of the polynomials in a minimal generating set that does not depend on $p$ when the dimension of the representation is sufficiently large. We also show that $p+1$ is the minimal number such that the invariants up to that degree always form a separating set. As well, we give an explicit description of a separating set when $p$ is prime.Comment: 7 page