21,165 research outputs found
The most and the least avoided consecutive patterns
We prove that the number of permutations avoiding an arbitrary consecutive
pattern of length m is asymptotically largest when the avoided pattern is
12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles
a conjecture of the author and Noy from 2001, as well as another recent
conjecture of Nakamura. We also show that among non-overlapping patterns of
length m, the pattern 134...m2 is the one for which the number of permutations
avoiding it is asymptotically largest
Linearised Higher Variational Equations
This work explores the tensor and combinatorial constructs underlying the
linearised higher-order variational equations of a generic autonomous system
along a particular solution. The main result of this paper is a compact yet
explicit and computationally amenable form for said variational systems and
their monodromy matrices. Alternatively, the same methods are useful to
retrieve, and sometimes simplify, systems satisfied by the coefficients of the
Taylor expansion of a formal first integral for a given dynamical system. This
is done in preparation for further results within Ziglin-Morales-Ramis theory,
specifically those of a constructive nature.Comment: Minor changes with respect to previous versio
Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian
This is an example of application of Ziglin-Morales-Ramis algebraic studies
in Hamiltonian integrability, more specifically the result by Morales, Ramis
and Sim\'o on higher-order variational equations, to the well-known
Friedmann-Robertson-Walker cosmological model. A previous paper by the author
formalises said variational systems in such a way allowing the simple
expression of notable elements of the differential Galois group needed to study
integrability. Using this formalisation and an alternative method already used
by other authors, we find sufficient conditions whose fulfillment would entail
very simple proofs of non-integrability -- both for the complete Hamiltonian, a
goal already achieved by other means by Coelho et al, and for a special open
case attracting recent attention.Comment: 11 Figures, 15 pages, changed title from previous versio
Computer Simulation of Quantum Dynamics in a Classical Spin Environment
In this paper a formalism for studying the dynamics of quantum systems
coupled to classical spin environments is reviewed. The theory is based on
generalized antisymmetric brackets and naturally predicts open-path
off-diagonal geometric phases in the evolution of the density matrix. It is
shown that such geometric phases must also be considered in the
quantum-classical Liouville equation for a classical bath with canonical phase
space coordinates; this occurs whenever the adiabatics basis is complex (as in
the case of a magnetic field coupled to the quantum subsystem). When the
quantum subsystem is weakly coupled to the spin environment, non-adiabatic
transitions can be neglected and one can construct an effective non-Markovian
computer simulation scheme for open quantum system dynamics in classical spin
environments. In order to tackle this case, integration algorithms based on the
symmetric Trotter factorization of the classical-like spin propagator are
derived. Such algorithms are applied to a model comprising a quantum two-level
system coupled to a single classical spin in an external magnetic field.
Starting from an excited state, the population difference and the coherences of
this two-state model are simulated in time while the dynamics of the classical
spin is monitored in detail. It is the author's opinion that the numerical
evidence provided in this paper is a first step toward developing the
simulation of quantum dynamics in classical spin environments into an effective
tool. In turn, the ability to simulate such a dynamics can have a positive
impact on various fields, among which, for example, nano-science.Comment: To appear in Theoretical Chemistry Accounts (special issue in honor
of Professor Gregory Sion Ezra
Generating trees for permutations avoiding generalized patterns
We construct generating trees with one, two, and three labels for some
classes of permutations avoiding generalized patterns of length 3 and 4. These
trees are built by adding at each level an entry to the right end of the
permutation, which allows us to incorporate the adjacency condition about some
entries in an occurrence of a generalized pattern. We use these trees to find
functional equations for the generating functions enumerating these classes of
permutations with respect to different parameters. In several cases we solve
them using the kernel method and some ideas of Bousquet-M\'elou. We obtain
refinements of known enumerative results and find new ones.Comment: 17 pages, to appear in Ann. Com
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