828 research outputs found
Surfaces Meeting Porous Sets in Positive Measure
Let n>2 and X be a Banach space of dimension strictly greater than n. We show
there exists a directionally porous set P in X for which the set of C^1
surfaces of dimension n meeting P in positive measure is not meager. If X is
separable this leads to a decomposition of X into a countable union of
directionally porous sets and a set which is null on residually many C^1
surfaces of dimension n. This is of interest in the study of certain classes of
null sets used to investigate differentiability of Lipschitz functions on
Banach spaces
Amenability of algebras of approximable operators
We give a necessary and sufficient condition for amenability of the Banach
algebra of approximable operators on a Banach space. We further investigate the
relationship between amenability of this algebra and factorization of
operators, strengthening known results and developing new techniques to
determine whether or not a given Banach space carries an amenable algebra of
approximable operators. Using these techniques, we are able to show, among
other things, the non-amenability of the algebra of approximable operators on
Tsirelson's space.Comment: 20 pages, to appear in Israel Journal of Mathematic
Genericity of Fr\'echet smooth spaces
If a separable Banach space contains an isometric copy of every separable
reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of
every separable Banach space. The same conclusion holds if we consider
separable Banach spaces with Fr\'echet smooth dual space. This improves a
result of G. Godefroy and N. J. Kalton.Comment: 34 page
Fields Medals and Nevanlinna Prize Presented at ICM-94 in Zurich
The Notices solicited the following five articles describing the work of the Fields Medalists and Nevanlinna Prize winner
Selective amplification of scars in a chaotic optical fiber
In this letter we propose an original mechanism to select scar modes through
coherent gain amplification in a multimode D-shaped fiber. More precisely, we
numerically demonstrate how scar modes can be amplified by positioning a gain
region in the vicinity of specific points of a short periodic orbit known to
give rise to scar modes
Bounding the dimension of bipartite quantum systems
Let us consider the set of joint quantum correlations arising from
two-outcome local measurements on a bipartite quantum system. We prove that no
finite dimension is sufficient to generate all these sets. We approach the
problem in two different ways by constructing explicit examples for every
dimension d, which demonstrates that there exist bipartite correlations that
necessitate d-dimensional local quantum systems in order to generate them. We
also show that at least 10 two-outcome measurements must be carried out by the
two parties altogether so as to generate bipartite joint correlations not
achievable by two-dimensional local systems. The smallest explicit example we
found involves 11 settings.Comment: 9 pages, no figures; published versio
Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Carathéodory's Theorem
We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in R^d, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result.
Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time n^O(log s/ε^2}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003).
The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time n^O(log d/ε^2)
Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms
We exhibit scarring for certain nonlinear ergodic toral automorphisms. There
are perturbed quantized hyperbolic toral automorphisms preserving certain
co-isotropic submanifolds. The classical dynamics is ergodic, hence in the
semiclassical limit almost all eigenstates converge to the volume measure of
the torus. Nevertheless, we show that for each of the invariant submanifolds,
there are also eigenstates which localize and converge to the volume measure of
the corresponding submanifold.Comment: 17 page
Quantum site percolation on amenable graphs
We consider the quantum site percolation model on graphs with an amenable
group action. It consists of a random family of Hamiltonians. Basic spectral
properties of these operators are derived: non-randomness of the spectrum and
its components, existence of an self-averaging integrated density of states and
an associated trace-formula.Comment: 10 pages, LaTeX 2e, to appear in "Applied Mathematics and Scientific
Computing", Brijuni, June 23-27, 2003. by Kluwer publisher
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