Electronic Research Announcements

Open Access Articles

Fixed frequency eigenfunction immersions and supremum norms of random waves
Yaiza Canzani and Boris Hanin
2015, 22: 76-86 doi: 10.3934/era.2015.22.76 +[Abstract](2898) +[PDF](381.3KB)
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 Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.
Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result
Gabriella Pinzari
2015, 22: 55-75 doi: 10.3934/era.2015.22.55 +[Abstract](3212) +[PDF](549.2KB)
We improve a result in [9] by proving the existence of a positive measure set of $(3n-2)$-dimensional quasi-periodic motions in the spacial, planetary $(1+n)$-body problem away from co-planar, circular motions. We also prove that such quasi-periodic motions reach with continuity corresponding $(2n-1)$-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 quasi-integrable structure, with a small remainder, independently of eccentricities and inclinations.
A sharp Sobolev-Strichartz estimate for the wave equation
Neal Bez and Chris Jeavons
2015, 22: 46-54 doi: 10.3934/era.2015.22.46 +[Abstract](3138) +[PDF](350.4KB)
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 $L^4$ Sobolev--Strichartz estimate for the wave equation in four spatial dimensions.
Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity
Jianwei Yang, Peng Cheng and Yudong Wang
2015, 22: 20-31 doi: 10.3934/era.2015.22.20 +[Abstract](3012) +[PDF](373.6KB)
In this paper, we study a combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes-Korteweg equations for weak solutions. For well prepared initial data, the convergence of solutions of the compressible Navier-Stokes-Korteweg equations to the solutions of the incompressible Navier-Stokes equation are justified rigorously by adapting the modulated energy method. Furthermore, the corresponding convergence rates are also obtained.
The $\boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem
Rüdiger Achilles, Andrea Bonfiglioli and Jacob Katriel
2015, 22: 32-45 doi: 10.3934/era.2015.22.32 +[Abstract](2540) +[PDF](447.5KB)
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: $[n]_q!:=[n-1]_q!\cdot(q^n-1)/(q-1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are non-commuting indeterminates and $[y,x]_q$ is the $q$-commutator $yx-q\,xy$, 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 well-known 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 $Q_{i,j}$.
Smoothing 3-dimensional polyhedral spaces
Nina Lebedeva, Vladimir Matveev, Anton Petrunin and Vsevolod Shevchishin
2015, 22: 12-19 doi: 10.3934/era.2015.22.12 +[Abstract](2466) +[PDF](324.6KB)
We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.
The approximate Loebl-Komlós-Sós conjecture and embedding trees in sparse graphs
Jan Hladký, Diana Piguet, Miklós Simonovits, Maya Stein and Endre Szemerédi
2015, 22: 1-11 doi: 10.3934/era.2015.22.1 +[Abstract](3570) +[PDF](423.3KB)
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].
An inverse theorem for the Gowers $U^{s+1}[N]$-norm
Ben Green, Terence Tao and Tamar Ziegler
2011, 18: 69-90 doi: 10.3934/era.2011.18.69 +[Abstract](5533) +[PDF](308.3KB)
This is an announcement of the proof of the inverse conjecture for the Gowers $U^{s+1}[N]$-norm for all $s \geq 3$; this is new for $s \geq 4$, the cases $s = 1,2,3$ having been previously established. More precisely we outline a proof that if $f : [N] \rightarrow [-1,1]$ is a function with ||$f$|| $U^{s+1}[N] \geq \delta$ then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ which correlates with $f$, where the bounds on the complexity and correlation depend only on $s$ and $\delta$. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. In particular, one obtains an asymptotic formula for the number of $k$-term arithmetic progressions $p_1 < p_2 < ... < p_k \leq N$ of primes, for every $k \geq 3$.
Linear approximate groups
Emmanuel Breuillard, Ben Green and Terence Tao
2010, 17: 57-67 doi: 10.3934/era.2010.17.57 +[Abstract](3451) +[PDF](202.0KB)
This is an informal announcement of results to be described and proved in detail in [3]. We give various results on the structure of approximate subgroups in linear groups such as ${\rm{S}}{{\rm{L}}_n}(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of ${\rm{S}}{{\rm{L}}_n}({\mathbb{F}_q})$ which generates the group must be either very small or else nearly all of ${\rm{S}}{{\rm{L}}_n}({\mathbb{F}_q})$. The argument is valid for all Chevalley groups $G(\mathbb{F}_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in [4].
Pointwise theorems for amenable groups
Elon Lindenstrauss
1999, 5: 82-90 +[Abstract](1039) +[PDF](207.5KB)

2020 Impact Factor: 0.929
5 Year Impact Factor: 0.674





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