On the spectral gap of some Cayley graphs on the Weyl group W(Bn)W(B_n)

The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N = 2^n n!$ equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a $2n \times 2n$ matrix associated to a $2n$-dimensional permutation representation of $W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous' spectral gap conjecture', proven in 2010 by Caputo, Liggett and Richthammer.Comment: Version 1 (v1) contains a mistake. The main result is proved here under a less general hypothesis than in v1. Main result of v1 is left as a conjectur

Public investment and higher education inequality

Empirical results show that children from high income households achieve higher levels of education and are more likely to be enrolled in post compulsory school. Theoretical findings fail to answer clearly whether greater public investment in the higher education system effectively decreases the inequality between the educational attainment of rich and poor children. We show that if the child receives a monetary transfer from his parents and allocates it between private consumption and investment in private additional education, then a further public investment decreases the educational gap. This result holds under the assumptions of both sub-stitutability and complementarity between private and public education.Higher education inequality; Public education; Altruism

Mergers under endogenous minimum quality standard: a note

We introduce merging strategies and endogenous MQS, borrowed from Ecchia and Lambertini (1997), in Scarpa (1998). MQS induces the low-quality firm to exit the market and leads to a monopoly arising from the bilateral merger of the high-quality firmsMergers, Minimum quality standard, Quality differentiation.

Profitability of Horizontal Mergers in Trigger Strategy Game

It is shown that, in a dynamic competition, an exogenous horizontal merger is profitable even if a small share of active firms merge. However, each firm has incentive to remain outside the merger because it would benefit more (Insiders’dilemma). We show that in an infinite repeated game in which the firms use trigger strategies an exogenous bilateral merger can be profitable and the Insiders’dilemma is mitigated.Horizontal mergers; Insiders’ dilemma; trigger strategy

Do childless households support local public provision of education

Empirical and theoretical studies show that the local provision of public education affects the well being of individuals through two channels: the first reflects the direct use of the good, whereas the second runs through the value of the housing. The second effect leans on the idea that the quality of public education is capitalized into the value of the own housing. Empirical evidence finds that in a multi-community model childless households support local public spending in education because of the capitalization effect. I study the behavior of childless households, not necessarily elderly, in a two community model and show that the capitalization effect may not be a sufficient condition for middle aged households without children to support local public spending in education by a majority voting.

Local public education and childless voting : the arising of an "ends with the middle" coalition

We show that capitalization of local education into the housing price induces childless voters to support local education. In particular, low income childless households vote for a tax raise when capitalization is strong, whereas high income childless supports a higher tax when capitalization is weak. The median income voter is never pivotal because "ends with the middle" coalitions arise: high income households (with and without a child) makes coalition with middle income class with a child, whereas low income households (with and without a child) make coalition with childless middle income class. We …nd that the income of the childless median voter is higher than the median income, whereas median voter with a child has income lower than the median. Thus the equilibrium tax preferred by the median voter (childless or not), is higher than the tax preferred by the childless median income voter and lower than the tax preferred by the median income voter with a child. This result implies that it is not possible to exclude voting equilibria in which thelocal public education; housings; local tax; capitalization; childless voting

Mergers under endogenous minimum quality standard: a note

We introduce merging strategies and endogenous MQS, borrowed from Ecchia and Lambertini (1997), in Scarpa (1998). MQS induces the low-quality firm to exit the market and leads to a monopoly arising from the bilateral merger of the high-quality firm

Sitopia

In the projected future urban world where NAFTA doesn\u27t exist, fresh/organic fruits and vegetables are produced within the urban realm. The goal is to create a symbiotic relationship between agriculture and architecture; analyze how spaces and typologies are challenged in order to accommodate farming; and systematize new spaces through restructuring of the code

Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap

In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of the symmetric group.Comment: Shorter version. Published in the Electron. J. of Combinatoric