4,913 research outputs found
On and Off-diagonal Sturmian operator: dynamic and spectral dimension
We study two versions of quasicrystal model, both subcases of Jacobi
matrices. For Off-diagonal model, we show an upper bound of dynamical exponent
and the norm of the transfer matrix. We apply this result to the Off-diagonal
Fibonacci Hamiltonian and obtain a sub-ballistic bound for coupling large
enough. In diagonal case, we improve previous lower bounds on the fractal
box-counting dimension of the spectrum.Comment: arXiv admin note: text overlap with arXiv:math-ph/0502044 and
arXiv:0807.3024 by other author
Spectral properties of the renormalization group at infinite temperature
The renormalization group (RG) approach is largely responsible for the
considerable success that has been achieved in developing a quantitative theory
of phase transitions. Physical properties emerge from spectral properties of
the linearization of the RG map at a fixed point. This article considers RG for
classical Ising-type lattice systems. The linearization acts on an
infinite-dimensional Banach space of interactions. At a trivial fixed point
(zero interaction), the spectral properties of the RG linearization can be
worked out explicitly, without any approximation. The results are for the RG
maps corresponding to decimation and majority rule. They indicate spectrum of
an unusual kind: dense point spectrum for which the adjoint operators have no
point spectrum at all, only residual spectrum. This may serve as a lesson in
what one might expect in more general situations.Comment: 12 page
Local quasi hidden variable modelling and violations of Bell-type inequalities by a multipartite quantum state
We introduce for a general correlation scenario a new simulation model, a
local quasi hidden variable (LqHV) model, where locality and the
measure-theoretic structure inherent to an LHV model are preserved but
positivity of a simulation measure is dropped. We specify a necessary and
sufficient condition for LqHV modelling and, based on this, prove that every
quantum correlation scenario admits an LqHV simulation. Via the LqHV approach,
we construct analogs of Bell-type inequalities for an N-partite quantum state
and find a new analytical upper bound on the maximal violation by an N-partite
quantum state of S_{1}x...xS_{N}-setting Bell-type inequalities - either on
correlation functions or on joint probabilities and for outcomes of an
arbitrary spectral type, discrete or continuous. This general analytical upper
bound is expressed in terms of the new state dilation characteristics
introduced in the present paper and not only traces quantum states admitting an
S_{1}x...xS_{N}-setting LHV description but also leads to the new exact
numerical upper estimates on the maximal Bell violations for concrete N-partite
quantum states used in quantum information processing and for an arbitrary
N-partite quantum state. We, in particular, prove that violation by an
N-partite quantum state of an arbitrary Bell-type inequality (either on
correlation functions or on joint probabilities) for S settings per site cannot
exceed (2S-1)^{N-1} even in case of an infinite dimensional quantum state and
infinitely many outcomes.Comment: Improved, edited versio
Generalized Solutions for Quantum Mechanical Oscillator on K\"{a}hler Conifold
We study the possible generalized boundary conditions and the corresponding
solutions for the quantum mechanical oscillator model on K\"{a}hler conifold.
We perform it by self-adjoint extension of the the initial domain of the
effective radial Hamiltonian. Remarkable effect of this generalized boundary
condition is that at certain boundary condition the orbital angular momentum
degeneracy is restored! We also recover the known spectrum in our formulation,
which of course correspond to some other boundary condition.Comment: 7 pages, latex, no figur
A rigorous approach to the magnetic response in disordered systems
This paper is a part of an ongoing study on the diamagnetic behavior of a
3-dimensional quantum gas of non-interacting charged particles subjected to an
external uniform magnetic field together with a random electric potential. We
prove the existence of an almost-sure non-random thermodynamic limit for the
grand-canonical pressure, magnetization and zero- field orbital magnetic
susceptibility. We also give an explicit formulation of these thermodynamic
limits. Our results cover a wide class of physically relevant random potentials
which model not only crystalline disordered solids, but also amorphous solids.Comment: 35 pages. Revised version. Accepted for publication in RM
Fr\'echet frames, general definition and expansions
We define an {\it -frame} with Banach spaces , , and a -space (\Theta, \snorm[\cdot]).
Then by the use of decreasing sequences of Banach spaces
and of sequence spaces , we define a general Fr\'
echet frame on the Fr\' echet space . We give
frame expansions of elements of and its dual , as well of some of
the generating spaces of with convergence in appropriate norms. Moreover,
we give necessary and sufficient conditions for a general pre-Fr\' echet frame
to be a general Fr\' echet frame, as well as for the complementedness of the
range of the analysis operator .Comment: A new section is added and a minor revision is don
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Teaching schools evaluation. Research Brief
This Research Brief reports the findings from a two-year study (2013-15) in to the work of teaching schools and their alliances commissioned by the National College for Teaching and Leadership (NCTL). The broad aim of the study was to investigate the effectiveness and impact of teaching schools on improvement, and identify the quality and scope of external support that are required to enhance these . This was achieved through combining qualitative and quantitative data collection and analysis derived from three research activities: case studies of 26 teaching schools alliances (TSAs), a national survey of the first three cohorts of 345 TSAs, and secondary research and analysis of national performance and inspection results
Analysis of unbounded operators and random motion
We study infinite weighted graphs with view to \textquotedblleft limits at
infinity,\textquotedblright or boundaries at infinity. Examples of such
weighted graphs arise in infinite (in practice, that means \textquotedblleft
very\textquotedblright large) networks of resistors, or in statistical
mechanics models for classical or quantum systems. But more generally our
analysis includes reproducing kernel Hilbert spaces and associated operators on
them. If is some infinite set of vertices or nodes, in applications the
essential ingredient going into the definition is a reproducing kernel Hilbert
space; it measures the differences of functions on evaluated on pairs of
points in . And the Hilbert norm-squared in will represent
a suitable measure of energy. Associated unbounded operators will define a
notion or dissipation, it can be a graph Laplacian, or a more abstract
unbounded Hermitian operator defined from the reproducing kernel Hilbert space
under study. We prove that there are two closed subspaces in reproducing kernel
Hilbert space which measure quantitative notions of limits at
infinity in , one generalizes finite-energy harmonic functions in
, and the other a deficiency index of a natural operator in
associated directly with the diffusion. We establish these
results in the abstract, and we offer examples and applications. Our results
are related to, but different from, potential theoretic notions of
\textquotedblleft boundaries\textquotedblright in more standard random walk
models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure
Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment
In this paper, we show that the concept of sigma-convergence associated to
stochastic processes can tackle the homogenization of stochastic partial
differential equations. In this regard, the homogenization problem for a
stochastic nonlinear partial differential equation is studied. Using some deep
compactness results such as the Prokhorov and Skorokhod theorems, we prove that
the sequence of solutions of this problem converges in probability towards the
solution of an equation of the same type. To proceed with, we use a suitable
version of sigma-convergence method, the sigma-convergence for stochastic
processes, which takes into account both the deterministic and random
behaviours of the solutions of the problem. We apply the homogenization result
to some concrete physical situations such as the periodicity, the almost
periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application
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