Electronic Research Announcements

2011 , Volume 18

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Special functions created by Borel-Laplace transform of Hénon map
Chihiro Matsuoka and  Koichi Hiraide
2011, 18: 1-11 doi: 10.3934/era.2011.18.1 +[Abstract](46) +[PDF](670.0KB)
We present a novel class of functions that can describe the stable and unstable manifolds of the Hénon map. We propose an algorithm to construct these functions by using the Borel-Laplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Hénon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.
On subgroups of the Dixmier group and Calogero-Moser spaces
Yuri Berest , Alimjon Eshmatov and  Farkhod Eshmatov
2011, 18: 12-21 doi: 10.3934/era.2011.18.12 +[Abstract](38) +[PDF](383.8KB)
We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1(k) $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key rôle in our approach is played by a transitive action of the automorphism group of the free algebra $ k< x, y>$ on the Calogero-Moser varieties $ \CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture for $ A_1(k) $ to the class of Morita equivalent algebras.
Realization of joint spectral radius via Ergodic theory
Xiongping Dai , Yu Huang and  Mingqing Xiao
2011, 18: 22-30 doi: 10.3934/era.2011.18.22 +[Abstract](96) +[PDF](181.2KB)
Based on the classic multiplicative ergodic theorem and the semi-uniform subadditive ergodic theorem, we show that there always exists at least one ergodic Borel probability measure such that the joint spectral radius of a finite set of square matrices of the same size can be realized almost everywhere with respect to this Borel probability measure. The existence of at least one ergodic Borel probability measure, in the context of the joint spectral radius problem, is obtained in a general setting.
Jordan elements and Left-Center of a Free Leibniz algebra
A. S. Dzhumadil'daev
2011, 18: 31-49 doi: 10.3934/era.2011.18.31 +[Abstract](46) +[PDF](201.1KB)
An element of a free Leibniz algebra is called Jordan if it belongs to a free Leibniz-Jordan subalgebra. Elements of the Jordan commutant of a free Leibniz algebra are called weak Jordan. We prove that an element of a free Leibniz algebra over a field of characteristic 0 is weak Jordan if and only if it is left-central. We show that free Leibniz algebra is an extension of a free Lie algebra by left-center. We find the dimensions of the homogeneous components of the Jordan commutant and the base of its multilinear part. We find criterion for an element of free Leibniz algebra to be Jordan.
Spectrum of some triangulated categories
Umesh V. Dubey and  Vivek M. Mallick
2011, 18: 50-53 doi: 10.3934/era.2011.18.50 +[Abstract](45) +[PDF](102.4KB)
In this note we announce the computation of the triangular spectrum (as defined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of $G$-equivariant sheaves on a smooth quasi projective scheme $X$ for a finite group $G$ which acts on $X$. The other class is the derived category of split super-schemes.
Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$
Hermann Köenig and  Vitali Milman
2011, 18: 54-60 doi: 10.3934/era.2011.18.54 +[Abstract](41) +[PDF](134.0KB)
Let $T:C^1(RR)\to C(RR)$ be an operator satisfying the derivation equation

$T(f\cdot g)=(Tf)\cdot g + f \cdot (Tg),$

where $f,g\in C^1(RR)$, and some weak additional assumption. Then $T$ must be of the form

$(Tf)(x) = c(x) \, f'(x) + d(x) \, f(x) \, \ln |f(x)|$

for $f \in C^1(RR), x \in RR$, where $c, d \in C(RR)$ are suitable continuous functions, with the convention $0 \ln 0 = 0$. If the domain of $T$ is assumed to be $C(RR)$, then $c=0$ and $T$ is essentially given by the entropy function $f \ln |f|$. We can also determine the solutions of the generalized derivation equation

$T(f\cdot g)=(Tf)\cdot (A_1g) + (A_2f) \cdot (Tg), $

where $f,g\in C^1(RR)$, for operators $T:C^1(RR)\to C(RR)$ and $A_1, A_2:C(RR)\to C(RR)$ fulfilling some weak additional properties.

Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets
Anat Amir
2011, 18: 61-68 doi: 10.3934/era.2011.18.61 +[Abstract](36) +[PDF](142.2KB)
Zapolsky's inequality gives a lower bound for the $L_1$ norm of the Poisson bracket of a pair of $C^1$ functions on the two-dimensional sphere by means of quasi-states. Here we show that this lower bound is sharp.
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](88) +[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 $k \geq 3$.
Deligne pairing and determinant bundle
Indranil Biswas , Georg Schumacher and  Lin Weng
2011, 18: 91-96 doi: 10.3934/era.2011.18.91 +[Abstract](48) +[PDF](134.9KB)
Let $X \rightarrow\ S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L_0\, L_1\, \cdots \, L_{n-1}\, L_{n}$ be line bundles over $X$. There is a natural isomorphism of the Deligne pairing 〈$L_0\, \cdots\, L_{n}$〉with the determinant line bundle $Det(\otimes_{i=0}^{n} (L_i- \mathcal O_{X}))$.
Simple loops on 2-bridge spheres in 2-bridge link complements
Donghi Lee and  Makoto Sakuma
2011, 18: 97-111 doi: 10.3934/era.2011.18.97 +[Abstract](34) +[PDF](447.7KB)
The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane's identity.
Order isomorphisms in windows
Shiri Artstein-Avidan , Dan Florentin and  Vitali Milman
2011, 18: 112-118 doi: 10.3934/era.2011.18.112 +[Abstract](112) +[PDF](142.8KB)
We characterize order preserving transforms on the class of lower-semi-continuous convex functions that are defined on a convex subset of $\mathbb{R}^n$ (a "window") and some of its variants. To this end, we investigate convexity preserving maps on subsets of $\mathbb{R}^n$. We prove that, in general, an order isomorphism is induced by a special convexity preserving point map on the epi-graph of the function. In the case of non-negative convex functions on $K$, where $0\in K$ and $f(0) = 0$, one may naturally partition the set of order isomorphisms into two classes; we explain the main ideas behind these results.
Equivariant sheaves on some spherical varieties
Aravind Asok and  James Parson
2011, 18: 119-130 doi: 10.3934/era.2011.18.119 +[Abstract](33) +[PDF](201.2KB)
We provide a concrete description of the category of equivariant vector bundles on a class of spherical $\G$-varieties.
Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes
Maksim Maydanskiy and  Benjamin P. Mirabelli
2011, 18: 131-143 doi: 10.3934/era.2011.18.131 +[Abstract](49) +[PDF](376.7KB)
The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, $W$, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of $W$ are non-degenerate. In this paper, we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.

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