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Free to readers and authors, Electronic Research Announcements in Mathematical Sciences rapidly publishes announcements of significant advances in all branches of mathematics and short complete papers of original research (up to about 15 journal pages). Research announcements are an opportunity for lucid exposition of ideas and context unburdened by technical detail. All articles should be designed to communicate their contents to a broad mathematical audience and must meet high standards for mathematical content and clarity. After review and acceptance by the entire Editorial Board, articles enter production for immediate publication.
ERA is a continuation of Electronic Research Announcements of the AMS published by the American Mathematical Society from 1995 to the middle of 2007. After over two decades of leading this journal, Svetlana Katok became Founding Editor Emerita in January 2017.
 AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
 Publishes 1 issue a year.
 Publishes online only.
 Archived in Portico and CLOCKSS.
 ERAMS is a publication of the American Institute of Mathematical Sciences. All rights reserved.

TOP 10 Most Read Articles in ERAMS, September 2017
1 
Research announcement: The structure of groups with a quasiconvex hierarchy
Volume 16, Number 0, Pages: 44  55, 2009
Daniel T. Wise
Abstract
Full Text
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Let $G$ be a wordhyperbolic group with a quasiconvex hierarchy.
We show that $G$ has a finite index subgroup $G'$ that embeds as a
quasiconvex subgroup of a rightangled Artin group.
It follows that every quasiconvex subgroup of $G$ is a virtual retract,
and is hence separable.
The results are applied to certain 3manifold and onerelator groups.

2 
Unboundedness of the Lagrangian Hofer distance in the Euclidean ball
Volume 21, Number 0, Pages: 1  7, 2014
Sobhan Seyfaddini
Abstract
References
Full Text
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Let $\mathcal{L}$ denote the space of Lagrangians Hamiltonian isotopic to the standard Lagrangian in the unit ball in $\mathbb{R}^{2n}$. We prove that the Lagrangian Hofer distance on $\mathcal{L}$ is unbounded.

3 
Theory of $(a,b)$continued fraction transformations and applications
Volume 17, Number 0, Pages: 20  33, 2010
Svetlana Katok
and Ilie Ugarcovici
Abstract
Full Text
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We study a twoparameter family of onedimensional maps and the related $(a,b)$continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps
have attractors with finite rectangular structure for the entire parameter set except for a Cantorlike set of onedimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations.
We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.

4 
The spectral gap of graphs and Steklov eigenvalues on surfaces
Volume 21, Number 0, Pages: 19  27, 2014
Bruno Colbois
and Alexandre Girouard
Abstract
References
Full Text
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Using expander graphs, we construct a sequence
$\{\Omega_N\}_{N\in\mathbb{N}}$ of smooth compact surfaces with boundary of
perimeter $N$, and with the first nonzero Steklov
eigenvalue $\sigma_1(\Omega_N)$ uniformly bounded away from
zero. This answers a question which was raised in [10]. The
sequence $\sigma_1(\Omega_N) L(\partial\Omega_n)$ grows linearly with the genus of
$\Omega_N$, which is the optimal growth rate.

5 
On degenerations of moduli of Hitchin pairs
Volume 20, Number 0, Pages: 103  108, 2013
V. Balaji,
P. Barik
and D. S. Nagaraj
Abstract
References
Full Text
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The purpose of this note is to announce certain basic results on the construction of a degeneration of ${\mathcal{M}}_{{{X_{k}}}}^{{H}}(n,d)$ as the smooth curve $X_{k}$ degenerates to an irreducible nodal curve with a single node.

6 
Optimally sparse 3D approximations using shearlet representations
Volume 17, Number 0, Pages: 125  137, 2010
Kanghui Guo
and Demetrio Labate
Abstract
References
Full Text
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This paper introduces a new Parseval frame, based on the 3D
shearlet representation, which is especially designed to capture
geometric features such as discontinuous boundaries with very high
efficiency. We show that this approach exhibits essentially optimal
approximation properties for 3D functions $f$ which are smooth
away from discontinuities along $C^2$ surfaces. In fact, the $N$
term approximation $f_N^S$ obtained by selecting the $N$ largest
coefficients from the shearlet expansion of $f$ satisfies the
asymptotic estimate
$ff_N^S$$_2^2$ ≍ $N^{1} (\log N)^2, as
N \to \infty.$
Up to the logarithmic factor,
this is the optimal behavior for functions in this class and
significantly outperforms wavelet approximations, which only yields
a $N^{1/2}$ rate. Indeed, the wavelet approximation rate was the
best published nonadaptive result so far and the result presented in
this paper is the first nonadaptive construction which is provably
optimal (up to a loglike factor) for this class of 3D data.
Our estimate is consistent with the corresponding
2D (essentially) optimally sparse approximation results obtained
by the authors using 2D shearlets and by Candès and Donoho using
curvelets.

7 
Multifractal formalism derived from thermodynamics for general dynamical systems
Volume 17, Number 0, Pages: 1  11, 2010
Vaughn Climenhaga
Abstract
Full Text
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We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T$: $ \RR\to \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. Applications include most previously known results, as well as some new ones.

8 
Integration of exact Courant algebroids
Volume 19, Number 0, Pages: 58  76, 2012
David LiBland
and Pavol Ševera
Abstract
References
Full Text
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In this paper, we describe an integration of exact Courant algebroids to symplectic 2groupoids,
and we show that the differentiation procedure from [32] inverts our integration.

9 
New progress in nonuniform measure and cocycle rigidity
Volume 15, Number 0, Pages: 79  92,
Abstract
Full Text
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10 
Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets
Volume 21, Number 0, Pages: 8  18, 2014
Steve Hofmann,
Dorina Mitrea,
Marius Mitrea
and Andrew J. Morris
Abstract
References
Full Text
Related Articles
We announce a local $T(b)$ theorem, an inductive scheme, and $L^p$ extrapolation
results for $L^2$ square function estimates related to the analysis of integral
operators that act on AhlforsDavid regular sets of arbitrary codimension in
ambient quasimetric spaces. The inductive scheme is a natural application of
the local $T(b)$ theorem and it implies the stability of $L^2$ square function
estimates under the socalled big pieces functor. In particular, this analysis
implies $L^p$ and Hardy space square function estimates for integral operators
on uniformly rectifiable subsets of the Euclidean space.

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