Electronic Research Announcements

January 2015 , Volume 22

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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](796) +[PDF](423.3KB)
Abstract:
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].
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](610) +[PDF](324.6KB)
Abstract:
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.
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](562) +[PDF](373.6KB)
Abstract:
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](521) +[PDF](447.5KB)
Abstract:
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 $Q_{i,j}$.
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](644) +[PDF](350.4KB)
Abstract:
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.
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](745) +[PDF](549.2KB)
Abstract:
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.
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](665) +[PDF](381.3KB)
Abstract:
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.
A Besicovitch cylindrical transformation with Hölder function
Andrey Kochergin
2015, 22: 87-91 doi: 10.3934/era.2015.22.87 +[Abstract](490) +[PDF](289.6KB)
Abstract:
For any $\gamma\in(0,1)$ and any $\varepsilon>0$ we construct a cylindrical cascade over some irrational circle rotation with a $\gamma$-Hölder function such that the Besicovitch condition holds and the Hausdorff dimension of the set of points in the circle having discrete orbits is more than $1-\gamma-\varepsilon$. This result gives the answers to some questions of K. Frączek and M. Lemańczyk [1].
Proof of the main conjecture on $g$-areas
François Lalonde and Egor Shelukhin
2015, 22: 92-102 doi: 10.3934/era.2015.22.92 +[Abstract](524) +[PDF](351.2KB)
Abstract:
In this paper, we prove the main conjecture on $g$-areas arising from [7]. That conjecture was announced by the first author in 2004. It~states that the $g$-area of any Hamiltonian diffeomorphism $\phi$ is equal to the positive Hofer pseudo-distance between $\phi$ and the subspace of Hamiltonian diffeomorphisms that can be expressed as a product of at most $g$ commutators.
On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces
B. Harbourne, P. Pokora and H. Tutaj-Gasińska
2015, 22: 103-108 doi: 10.3934/era.2015.22.103 +[Abstract](468) +[PDF](311.7KB)
Abstract:
In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be $(-1)$-curves, there are examples where this is not true.

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