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Free to all readers and authors, Electronic Research Announcements in
Mathematical Sciences is a continuation of Electronic
Research Announcements of the AMS published by the American Mathematical Society
from 1995 to the middle of 2007. ERA 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.

TOP 10 Most Read Articles in ERAMS, February 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 
The approximate LoeblKomlósSós conjecture and embedding trees in sparse graphs
Volume 22, Number 0, Pages: 1  11, 2015
Jan Hladký,
Diana Piguet,
Miklós Simonovits,
Maya Stein
and Endre Szemerédi
Abstract
References
Full Text
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Loebl, Komlós and Sós conjectured that every $n$vertex graph $G$ with at least $n/2$ vertices
of degree at least $k$ contains each tree $T$ of order $k+1$ as a
subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$.
For our proof, we use a structural decomposition which can be seen as an analogue of Szemerédi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$.
The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].

3 
Asymptotic limit of a NavierStokesKorteweg system with densitydependent viscosity
Volume 22, Number 0, Pages: 20  31, 2015
Jianwei Yang,
Peng Cheng
and Yudong Wang
Abstract
References
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In this paper, we study a combined incompressible and vanishing
capillarity limit in the barotropic compressible
NavierStokesKorteweg equations for weak solutions. For well
prepared initial data, the convergence of solutions of the
compressible NavierStokesKorteweg equations to the
solutions of the incompressible NavierStokes equation are justified
rigorously by adapting the modulated energy method. Furthermore, the
corresponding convergence rates are also obtained.

4 
Global Kolmogorov tori in the planetary $\boldsymbol N$body problem. Announcement of result
Volume 22, Number 0, Pages: 55  75, 2015
Gabriella Pinzari
Abstract
References
Full Text
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We improve a result in [9] by proving the existence of a positive measure set of $(3n2)$dimensional quasiperiodic motions in the spacial, planetary $(1+n)$body problem away from coplanar, circular motions. We also prove that such quasiperiodic motions reach with continuity corresponding $(2n1)$dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [2]).
The main tool is a full reduction of the SO(3)symmetry, which retains symmetry by reflections and highlights a quasiintegrable structure, with a small remainder, independently of eccentricities and inclinations.

5 
Smoothing 3dimensional polyhedral spaces
Volume 22, Number 0, Pages: 12  19, 2015
Nina Lebedeva,
Vladimir Matveev,
Anton Petrunin
and Vsevolod Shevchishin
Abstract
References
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We show that 3dimensional polyhedral manifolds
with nonnegative curvature in the sense of Alexandrov
can be approximated by nonnegatively curved 3dimensional Riemannian manifolds.

6 
The $\boldsymbol{q}$deformed CampbellBakerHausdorffDynkin theorem
Volume 22, Number 0, Pages: 32  45, 2015
Rüdiger Achilles,
Andrea Bonfiglioli
and Jacob Katriel
Abstract
References
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We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the $q$exponential $\exp_q(x)=\sum_{n=0}^{\infty} \frac{x^n}{[n]_q!}$, with the usual notation for $q$factorials: $[n]_q!:=[n1]_q!\cdot(q^n1)/(q1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are noncommuting indeterminates and $[y,x]_q$ is the $q$commutator $yxq\,xy$, then there exist linear combinations $Q_{i,j}(x,y)$ of iterated $q$commutators with exactly $i$ $x$'s, $j$ $y$'s and $[y,x]_q$ in their central position, such that $\exp_q(x)\exp_q(y)=\exp_q\!\big(x+y+\sum_{i,j\geq 1}Q_{i,j}(x,y)\big)$. Our expansion is consistent with the wellknown result by Schützenberger ensuring that one has $\exp_q(x)\exp_q(y)=\exp_q(x+y)$ if and only if $[y,x]_q=0$, and it improves former partial results on $q$deformed exponentiation. Furthermore, we give an algorithm which produces conjecturally a minimal generating set for the relations between $[y,x]_q$centered $q$commutators of any bidegree $(i,j)$, and it allows us to compute all possible $Q_{i,j}$.

7 
A sharp SobolevStrichartz estimate for the wave equation
Volume 22, Number 0, Pages: 46  54, 2015
Neal Bez
and Chris Jeavons
Abstract
References
Full Text
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We calculate the the sharp constant and characterize the extremal initial data in $\dot{H}^{\frac{3}{4}} \times \dot{H}^{\frac{1}{4}}$ for the $L^4$ SobolevStrichartz estimate for the wave equation in four spatial dimensions.

8 
Fixed frequency eigenfunction immersions and supremum norms of random waves
Volume 22, Number 0, Pages: 76  86, 2015
Yaiza Canzani
and Boris Hanin
Abstract
References
Full Text
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A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of BurqLebeau and others on upper bounds for the supnorms of random linear combinations of high frequency eigenfunctions.

9 
A characterization of the concept of duality
Volume 14, Number 0, Pages: 42  59,
Abstract
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10 
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.

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