Linear-time list recovery of high-rate expander codes

We show that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms. List recoverable codes have been useful recently in constructing efficiently list-decodable codes, as well as explicit constructions of matrices for compressive sensing and group testing. Previous list recoverable codes with linear-time decoding algorithms have all had rate at most 1/2; in contrast, our codes can have rate $1 - \epsilon$ for any $\epsilon > 0$. We can plug our high-rate codes into a construction of Meir (2014) to obtain linear-time list recoverable codes of arbitrary rates, which approach the optimal trade-off between the number of non-trivial lists provided and the rate of the code. While list-recovery is interesting on its own, our primary motivation is applications to list-decoding. A slight strengthening of our result would implies linear-time and optimally list-decodable codes for all rates, and our work is a step in the direction of solving this important problem

Superrigid subgroups and syndetic hulls in solvable Lie groups

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete subgroups D of connected Lie groups G, such that the homomorphisms defined on D can ("virtually") be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that D has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls.Comment: 17 pages. This is the final version that will appear in the volume "Rigidity in Dynamics and Geometry," edited by M. Burger and A. Iozzi (Springer, 2002

On encoding symbol degrees of array BP-XOR codes

Low density parity check (LDPC) codes, LT codes and digital fountain techniques have received significant attention from both academics and industry in the past few years. By employing the underlying ideas of efficient Belief Propagation (BP) decoding process (also called iterative message passing decoding process) on binary erasure channels (BEC) in LDPC codes, Wang has recently introduced the concept of array BP-XOR codes and showed the necessary and sufficient conditions for MDS [k + 2,k] and [n,2] array BP-XOR codes. In this paper, we analyze the encoding symbol degree requirements for array BP-XOR codes and present new necessary conditions for array BP-XOR codes. These new necessary conditions are used as a guideline for constructing several array BP-XOR codes and for presenting a complete characterization (necessary and sufficient conditions) of degree two array BP-XOR codes and for designing new edge-colored graphs. Meanwhile, these new necessary conditions are used to show that the codes by Feng, Deng, Bao, and Shen in IEEE Transactions on Computers are incorrect

Counting and effective rigidity in algebra and geometry

The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.Comment: v.2, 39 pages. To appear in Invent. Mat

Light States in Chern-Simons Theory Coupled to Fundamental Matter

Motivated by developments in vectorlike holography, we study SU(N) Chern-Simons theory coupled to matter fields in the fundamental representation on various spatial manifolds. On the spatial torus T^2, we find light states at small `t Hooft coupling \lambda=N/k, where k is the Chern-Simons level, taken to be large. In the free scalar theory the gaps are of order \sqrt {\lambda}/N and in the critical scalar theory and the free fermion theory they are of order \lambda/N. The entropy of these states grows like N Log(k). We briefly consider spatial surfaces of higher genus. Based on results from pure Chern-Simons theory, it appears that there are light states with entropy that grows even faster, like N^2 Log(k). This is consistent with the log of the partition function on the three sphere S^3, which also behaves like N^2 Log(k). These light states require bulk dynamics beyond standard Vasiliev higher spin gravity to explain them.Comment: 58 pages, LaTeX, no figures, Minor error corrected, references added, The main results of the paper have not change

Spacelike Singularities and Hidden Symmetries of Gravity

We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.Comment: 228 pages. Typos corrected. References added. Subject index added. Published versio

Amenability of groups and GG-sets

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups

Variation and ethnic inequalities in treatment of common mental disorders before, during and after pregnancy : combined analysis of routine and research data in the Born in Bradford cohort

BACKGROUND: Common mental disorders (CMD) such as anxiety and depression during the maternal period can cause significant morbidity to the mother in addition to disrupting biological, attachment and parenting processes that affect child development. Pharmacological treatment is a first-line option for moderate to severe episodes. Many women prescribed pharmacological treatments cease them during pregnancy but it is unclear to what extent non-pharmacological options are offered as replacement. There are also concerns that treatments offered may not be proportionate to need in minority ethnic groups, but few data exist on treatment disparities in the maternal period. We examined these questions in a multi-ethnic cohort of women with CMD living in Bradford, England before, during and up to one year after pregnancy. METHODS: We searched the primary care records of women enrolled in the Born in Bradford cohort for diagnoses, symptoms, signs ('identification'), referrals for treatment, non-pharmacological and pharmacological treatment and monitoring ('treatment') related to CMD. Records were linked with maternity data to classify women identified with a CMD as treated prior to, and one year after, delivery. We examined rates and types of treatment during pregnancy, and analysed potential ethnic group differences using adjusted Poisson and multinomial logistic regression models. RESULTS: We analysed data on 2,234 women with indicators of CMD. Most women were discontinued from pharmacological treatment early in pregnancy, but this was accompanied by recorded access to non-drug treatments in only 15 % at the time of delivery. Fewer minority ethnic women accessed treatments compared to White British women despite minority ethnic women being 55-70 % more likely than White British women to have been identified with anxiety in their medical record. CONCLUSIONS: Very few women who discontinued pharmacological treatment early in their pregnancy were offered other non-pharmacological treatments as replacement, and most appeared to complete their pregnancy untreated. Further investigation is warranted to replicate the finding that minority ethnic women are more likely to be identified as being anxious or having anxiety and understand what causes the variation in access to treatments

Convergence of measures on compactifications of locally symmetric spaces

We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space S=Γ∖G/K is compact. More precisely, given a sequence of homogeneous probability measures on S, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including S itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when G=SL3(R) and Γ=SL3(Z)

Cardiovascular risk associated with the use of glitazones, metformin and sufonylureas: meta-analysis of published observational studies

BACKGROUND: The results of observational studies evaluating and comparing the cardiovascular safety of glitazones, metformin and sufonylureas are inconsistent.To conduct and evaluate heterogeneity in a meta-analysis of observational studies on the risk of acute myocardial infarction (AMI) or stroke in patients with type 2 diabetes using non-insulin blood glucose–lowering drugs (NIBGLD). METHODS: We systematically identified and reviewed studies evaluating NIBGLD in patients with type 2 diabetes indexed in Medline, Embase, or the Cochrane Library that met prespecified criteria. The quality of included studies was assessed with the RTI item bank. Results were combined using fixed- and random-effects models, and the Higgins I(2) statistic was used to evaluate heterogeneity. Sensitivity analyses by study quality were conducted. RESULTS: The summary relative risk (sRR) (95 % CI) of AMI for rosiglitazone versus pioglitazone was 1.13 (1.04–1.24) [I(2) = 55 %]. In the sensitivity analysis, heterogeneity was reduced [I(2) = 16 %]. The sRR (95 % CI) of stroke for rosiglitazone versus pioglitazone was 1.18 (1.02–1.36) [I(2) = 42 %]. There was strong evidence of heterogeneity related to study quality in the comparisons of rosiglitazone versus metformin and rosiglitazone versus sulfonylureas (I(2) ≥ 70 %). The sRR (95 % CI) of AMI for sulfonylurea versus metformin was 1.24 (1.14–1.34) [I(2) = 41 %] and for pioglitazone versus metformin was 1.02 (0.75–1.38) [I(2) = 17 %]. Sensitivity analyses decreased heterogeneity in most comparisons. CONCLUSION/INTERPRETATION: Sulfonylureas increased the risk of AMI by 24 % compared with metformin; an imprecise point estimate indicated no difference in risk of AMI when comparing pioglitazone with metformin. The presence of heterogeneity precluded any conclusions on the other comparisons. The quality assessment was valuable in identifying methodological problems in the individual studies and for analysing potential sources of heterogeneity. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s12872-016-0187-5) contains supplementary material, which is available to authorized users