Singular Fermi Surfaces I. General Power Counting and Higher Dimensional Cases

We prove regularity properties of the self-energy, to all orders in perturbation theory, for systems with singular Fermi surfaces which contain Van Hove points where the gradient of the dispersion relation vanishes. In this paper, we show for spatial dimensions $d \ge 3$ that despite the Van Hove singularity, the overlapping loop bounds we proved together with E. Trubowitz for regular non--nested Fermi surfaces [J. Stat. Phys. 84 (1996) 1209] still hold, provided that the Fermi surface satisfies a no-nesting condition. This implies that for a fixed interacting Fermi surface, the self-energy is a continuously differentiable function of frequency and momentum, so that the quasiparticle weight and the Fermi velocity remain close to their values in the noninteracting system to all orders in perturbation theory. In a companion paper, we treat the more singular two-dimensional case.Comment: 48 pages LaTeX with figure

Perturbation Theory around Non-Nested Fermi Surfaces I. Keeping the Fermi Surface Fixed

The perturbation expansion for a general class of many-fermion systems with a non-nested, non-spherical Fermi surface is renormalized to all orders. In the limit as the infrared cutoff is removed, the counterterms converge to a finite limit which is differentiable in the band structure. The map from the renormalized to the bare band structure is shown to be locally injective. A new classification of graphs as overlapping or non-overlapping is given, and improved power counting bounds are derived from it. They imply that the only subgraphs that can generate $r$ factorials in the $r^{\rm th}$ order of the renormalized perturbation series are indeed the ladder graphs and thus give a precise sense to the statement that `ladders are the most divergent diagrams'. Our results apply directly to the Hubbard model at any filling except for half-filling. The half-filled Hubbard model is treated in another place.Comment: plain TeX with postscript figures in a uuencoded gz-compressed tar file. Put it on a separate directory before unpacking, since it contains about 40 files. If you have problems, requests or comments, send e-mail to [email protected]

Maximizing Symmetric Submodular Functions

Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little attention by current research, unlike similar minimization problems which have been widely studied. In this work, we identify a few submodular maximization problems for which one can get a better approximation for symmetric objectives than the state of the art approximation for general submodular functions. We first consider the problem of maximizing a non-negative symmetric submodular function $f\colon 2^\mathcal{N} \to \mathbb{R}^+$ subject to a down-monotone solvable polytope $\mathcal{P} \subseteq [0, 1]^\mathcal{N}$. For this problem we describe an algorithm producing a fractional solution of value at least $0.432 \cdot f(OPT)$, where $OPT$ is the optimal integral solution. Our second result considers the problem $\max \{f(S) : |S| = k\}$ for a non-negative symmetric submodular function $f\colon 2^\mathcal{N} \to \mathbb{R}^+$. For this problem, we give an approximation ratio that depends on the value $k / |\mathcal{N}|$ and is always at least $0.432$. Our method can also be applied to non-negative non-symmetric submodular functions, in which case it produces $1/e - o(1)$ approximation, improving over the best known result for this problem. For unconstrained maximization of a non-negative symmetric submodular function we describe a deterministic linear-time $1/2$-approximation algorithm. Finally, we give a $[1 - (1 - 1/k)^{k - 1}]$-approximation algorithm for Submodular Welfare with $k$ players having identical non-negative submodular utility functions, and show that this is the best possible approximation ratio for the problem.Comment: 31 pages, an extended abstract appeared in ESA 201

The Anderson Model as a Matrix Model

In this paper we describe a strategy to study the Anderson model of an electron in a random potential at weak coupling by a renormalization group analysis. There is an interesting technical analogy between this problem and the theory of random matrices. In d=2 the random matrices which appear are approximately of the free type well known to physicists and mathematicians, and their asymptotic eigenvalue distribution is therefore simply Wigner's law. However in d=3 the natural random matrices that appear have non-trivial constraints of a geometrical origin. It would be interesting to develop a general theory of these constrained random matrices, which presumably play an interesting role for many non-integrable problems related to diffusion. We present a first step in this direction, namely a rigorous bound on the tail of the eigenvalue distribution of such objects based on large deviation and graphical estimates. This bound allows to prove regularity and decay properties of the averaged Green's functions and the density of states for a three dimensional model with a thin conducting band and an energy close to the border of the band, for sufficiently small coupling constant.Comment: 23 pages, LateX, ps file available at http://cpth.polytechnique.fr/cpth/rivass/articles.htm

Migrants, immigrants and welfare from the Old Poor Law to the Welfare State

Under the Old Poor Law internal migrants moved from one jurisdiction to another when they crossed parochial boundaries. Following the Poor Law Amendment Act of 1834 central government took an enlarged and expanding part in welfare. As it did so, the entitlement to welfare of immigrants from overseas was scrutinised at a national level in a way that was analogous to the manner in which the status of internal migrants had previously been scrutinised at a parochial level. Having established this analogy, the essay asks whether the entitlement to welfare of outsiders improved or deteriorated over time and seeks to account for the broad trends

Learning DNF Expressions from Fourier Spectrum

Since its introduction by Valiant in 1984, PAC learning of DNF expressions remains one of the central problems in learning theory. We consider this problem in the setting where the underlying distribution is uniform, or more generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed that in this setting a DNF expression can be efficiently approximated from its "heavy" low-degree Fourier coefficients alone. This is in contrast to previous approaches where boosting was used and thus Fourier coefficients of the target function modified by various distributions were needed. This property is crucial for learning of DNF expressions over smoothed product distributions, a learning model introduced by Kalai et al. (2009) and inspired by the seminal smoothed analysis model of Spielman and Teng (2001). We introduce a new approach to learning (or approximating) a polynomial threshold functions which is based on creating a function with range [-1,1] that approximately agrees with the unknown function on low-degree Fourier coefficients. We then describe conditions under which this is sufficient for learning polynomial threshold functions. Our approach yields a new, simple algorithm for approximating any polynomial-size DNF expression from its "heavy" low-degree Fourier coefficients alone. Our algorithm greatly simplifies the proof of learnability of DNF expressions over smoothed product distributions. We also describe an application of our algorithm to learning monotone DNF expressions over product distributions. Building on the work of Servedio (2001), we give an algorithm that runs in time \poly((s \cdot \log{(s/\eps)})^{\log{(s/\eps)}}, n), where $s$ is the size of the target DNF expression and \eps is the accuracy. This improves on \poly((s \cdot \log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n) bound of Servedio (2001).Comment: Appears in Conference on Learning Theory (COLT) 201

Feminist science and epistemologies: Key issues central to GENNOVATE's research program

This methodological brief offers a window into GENNOVATE’s innovative collaborative research initiative to promote gender equality in agricultural and natural resource management. It addresses questions such as 1) Why is it important to distinguish among epistemology, methodology, and methods?; 2) What is feminist epistemology?; 3) What can researchers of gender, agriculture, and innovation learn from engaging the contributions of feminist epistemology?; and 4) How has GENNOVATE integrated lessons from feminist methods and feminist epistemics about gender relations, agricultural change, and innovation