16,440 research outputs found
Mean ergodicity vs weak almost periodicity
We provide explicit examples of positive and power-bounded operators on
and which are mean ergodic but not weakly almost periodic. As a
consequence we prove that a countably order complete Banach lattice on which
every positive and power-bounded mean ergodic operator is weakly almost
periodic is necessarily a KB-space. This answers several open questions from
the literature. Finally, we prove that if is a positive mean ergodic
operator with zero fixed space on an arbitrary Banach lattice, then so is every
power of .Comment: 10 pages; minor adjustments and three new references included
compared to version
Entanglement dynamics and quasi-periodicity in discrete quantum walks
We study the entanglement dynamics of discrete time quantum walks acting on
bounded finite sized graphs. We demonstrate that, depending on system
parameters, the dynamics may be monotonic, oscillatory but highly regular, or
quasi-periodic. While the dynamics of the system are not chaotic since the
system comprises linear evolution, the dynamics often exhibit some features
similar to chaos such as high sensitivity to the system's parameters,
irregularity and infinite periodicity. Our observations are of interest for
entanglement generation, which is one primary use for the quantum walk
formalism. Furthermore, we show that the systems we model can easily be mapped
to optical beamsplitter networks, rendering experimental observation of
quasi-periodic dynamics within reach.Comment: 9 pages, 8 figure
The Flux-Phase of the Half-Filled Band
The conjecture is verified that the optimum, energy minimizing magnetic flux
for a half-filled band of electrons hopping on a planar, bipartite graph is
per square plaquette. We require {\it only} that the graph has
periodicity in one direction and the result includes the hexagonal lattice
(with flux 0 per hexagon) as a special case. The theorem goes beyond previous
conjectures in several ways: (1) It does not assume, a-priori, that all
plaquettes have the same flux (as in Hofstadter's model); (2) A Hubbard type
on-site interaction of any sign, as well as certain longer range interactions,
can be included; (3) The conclusion holds for positive temperature as well as
the ground state; (4) The results hold in dimensions if there is
periodicity in directions (e.g., the cubic lattice has the lowest energy
if there is flux in each square face).Comment: 9 pages, EHL14/Aug/9
On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators
The article is devoted to the following question. Consider a periodic
self-adjoint difference (differential) operator on a graph (quantum graph) G
with a co-compact free action of the integer lattice Z^n. It is known that a
local perturbation of the operator might embed an eigenvalue into the
continuous spectrum (a feature uncommon for periodic elliptic operators of
second order). In all known constructions of such examples, the corresponding
eigenfunction is compactly supported. One wonders whether this must always be
the case. The paper answers this question affirmatively. What is more
surprising, one can estimate that the eigenmode must be localized not far away
from the perturbation (in a neighborhood of the perturbation's support, the
width of the neighborhood determined by the unperturbed operator only).
The validity of this result requires the condition of irreducibility of the
Fermi (Floquet) surface of the periodic operator, which is expected to be
satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio
Periodicity in Rank 2 Graph Algebras
Kumjian and Pask introduced an aperiodicity condition for higher rank graphs.
We present a detailed analysis of when this occurs in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof of the simplicity of
\ca(\Fth). The periodic C*-algebras are characterized, and it is shown that
\ca(\Fth) \simeq \rC(\bT) \otimes \fA where \fA is a simple C*-algebra.Comment: 27 page
On the location of spectral edges in -periodic media
Periodic nd order ordinary differential operators on are known to
have the edges of their spectra to occur only at the spectra of periodic and
antiperiodic boundary value problems. The multi-dimensional analog of this
property is false, as was shown in a 2007 paper by some of the authors of this
article. However, one sometimes encounters the claims that in the case of a
single periodicity (i.e., with respect to the lattice ), the
property still holds, and spectral edges occur at the periodic and
anti-periodic spectra only. In this work we show that even in the simplest case
of quantum graphs this is not true. It is shown that this is true if the graph
consists of a chain of finite graphs connected by single edges, while if
the connections are formed by at least two edges, the spectral edges can
already occur away from the periodic and anti-periodic spectra.Comment: 9 pages, 5 figure
Recompression: a simple and powerful technique for word equations
In this paper we present an application of a simple technique of local
recompression, previously developed by the author in the context of compressed
membership problems and compressed pattern matching, to word equations. The
technique is based on local modification of variables (replacing X by aX or Xa)
and iterative replacement of pairs of letters appearing in the equation by a
`fresh' letter, which can be seen as a bottom-up compression of the solution of
the given word equation, to be more specific, building an SLP (Straight-Line
Programme) for the solution of the word equation.
Using this technique we give a new, independent and self-contained proofs of
most of the known results for word equations. To be more specific, the
presented (nondeterministic) algorithm runs in O(n log n) space and in time
polynomial in log N, where N is the size of the length-minimal solution of the
word equation. The presented algorithm can be easily generalised to a generator
of all solutions of the given word equation (without increasing the space
usage). Furthermore, a further analysis of the algorithm yields a doubly
exponential upper bound on the size of the length-minimal solution. The
presented algorithm does not use exponential bound on the exponent of
periodicity. Conversely, the analysis of the algorithm yields an independent
proof of the exponential bound on exponent of periodicity.
We believe that the presented algorithm, its idea and analysis are far
simpler than all previously applied. Furthermore, thanks to it we can obtain a
unified and simple approach to most of known results for word equations.
As a small additional result we show that for O(1) variables (with arbitrary
many appearances in the equation) word equations can be solved in linear space,
i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were
simplified, overall presentation improve
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