Bounds for the Nakamura number

The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.Comment: 23 pages, 3 tables; a few more references adde

Bounds for the Nakamura number

This is a post-peer-review, pre-copyedit version of an article published in Social choice and welfare. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00355-018-1164-y.The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric (quota) games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending on invariants of simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.Peer ReviewedPostprint (author's final draft

Understanding the Random Displacement Model: From Ground-State Properties to Localization

We give a detailed survey of results obtained in the most recent half decade which led to a deeper understanding of the random displacement model, a model of a random Schr\"odinger operator which describes the quantum mechanics of an electron in a structurally disordered medium. These results started by identifying configurations which characterize minimal energy, then led to Lifshitz tail bounds on the integrated density of states as well as a Wegner estimate near the spectral minimum, which ultimately resulted in a proof of spectral and dynamical localization at low energy for the multi-dimensional random displacement model.Comment: 31 pages, 7 figures, final version, to appear in Proceedings of "Spectral Days 2010", Santiago, Chile, September 20-24, 201

The most and the least avoided consecutive patterns

We prove that the number of permutations avoiding an arbitrary consecutive pattern of length m is asymptotically largest when the avoided pattern is 12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles a conjecture of the author and Noy from 2001, as well as another recent conjecture of Nakamura. We also show that among non-overlapping patterns of length m, the pattern 134...m2 is the one for which the number of permutations avoiding it is asymptotically largest

Bounds on the spectral shift function and the density of states

We study spectra of Schr\"odinger operators on \RR^d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values $\mu_n$ of the difference of the semigroups as $n\to \infty$ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schr\"odinger operators. The single site potential $u$ is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be H\"older continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies H\"older continuity of the integrated density of states.Comment: Latex 2e, 15 page