506 research outputs found
Sharp spectral estimates for periodic matrix-valued Jacobi operators
For the periodic matrix-valued Jacobi operator we obtain the estimate of
the Lebesgue measure of the spectrum |\s(J)|\le4 \min_n\Tr(a_na_n^*)^\frac12,
where are off-diagonal elements of . Moreover estimates of width of
spectral bands are obtained.Comment: 3 page
Determinants and traces of multidimensional discrete periodic operators with defects
As it is shown in previous works, discrete periodic operators with defects
are unitarily equivalent to the operators of the form where
are continuous matrix-valued functions of appropriate sizes. All such operators
form a non-closed algebra . In this article we show that
there exist a trace and a determinant defined for
operators from with the properties The mappings
, are vector-valued functions. While has a
complex structure, is simple There exists the norm under which the closure
is a Banach algebra, and ,
are continuous (analytic) mappings. This algebra contains
simultaneously all operators of multiplication by matrix-valued functions and
all operators from the trace class. Thus, it generalizes the other algebras for
which determinants and traces was previously defined
Mixed multidimensional integral operators with piecewise constant kernels and their representations
We consider the algebra of mixed multidimensional integral operators. In
particular, Fredholm integral operators of the first and second kind belongs to
this algebra. For the piecewise constant kernels we provide an explicit
representation of the algebra as a product of simple matrix algebras. This
representation allows us to compute the inverse operators (or to solve the
corresponding integral equations) and to find the spectrum explicitly.
Moreover, explicit traces and determinants are also constructed. So, roughly
speaking, the analysis of integral operators is reduced to the analysis of
matrices. All the qualitative characteristics of the spectrum are preserved
since only the kernels are approximated
Application of matrix-valued integral continued fractions to spectral problems on periodic graphs
We show that spectral problems for periodic operators on lattices with
embedded defects of lower dimensions can be solved with the help of
matrix-valued integral continued fractions. While these continued fractions are
usual in the approximation theory, they are less known in the context of
spectral problems. We show that the spectral points can be expressed as zeroes
of determinants of the continued fractions. They are also useful in the study
of inverse problems (one-to-one correspondence between spectral data and
defects). Finally, the explicit formula for the resolvent in terms of the
continued fractions is also provided. We apply some of our results to the
Schr\"odinger operator acting on the graphene with line and point defects
Recovery of defects from the information at detectors
The discrete wave equation in a multidimensional uniform space with local
defects and sources is considered. The characterization of all possible defect
configurations corresponding to given amplitudes of waves at the receivers
(detectors) is provided.Comment: in Inverse Problems, 055005, 201
On the measure of the spectrum of direct integrals
We obtain the estimate of the Lebesgue measure of the spectrum for the direct
integral of matrix-valued functions. These estimates are applicable for a wide
class of discrete periodic operators. For example: these results give new and
sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic
Schrodinger operators
A note on sharp spectral estimates for periodic Jacobi matrices
The spectrum of three-diagonal self-adjoint -periodic Jacobi matrix with
positive off-diagonal elements an real diagonal elements consist of
intervals separated by gaps , where some of the gaps can be
degenerated. The following estimate is true We show that for any
there are Jacobi matrices of minimal period for which the spectral estimate
is sharp. The estimate is sharp for both: strongly and weakly oscillated ,
. Moreover, it improves some recent spectral estimates
Finite PDEs and finite ODEs are isomorphic
The standard view is that PDEs are much more complex than ODEs, but, as will
be shown below, for finite derivatives this is not true. We consider the
-algebras consisting of -dimensional finite
differential operators with -matrix-valued bounded periodic
coefficients. We show that any is -isomorphic to the
universal uniformly hyperfinite algebra (UHF algebra) This is a complete
characterization of the differential algebras. In particular, for different
the algebras are topologically and
algebraically isomorphic to each other. In this sense, there is no difference
between multidimensional matrix valued PDEs and
one-dimensional scalar ODEs . Roughly speaking, the
multidimensional world can be emulated by the one-dimensional one
Divergence of the logarithm of a unimodular monodromy matrix near the edges of the Brillouin zone
A first-order differential system with matrix of periodic coefficients
is studied for time-harmonic elastic waves in a unidirectionally
periodic medium, for which the monodromy matrix implies a
propagator of the wave field over a period. The main interest in the matrix
logarithm is due to the fact that it yields the 'effective'
matrix of the dynamic-homogenization method. For the typical
case of a unimodular matrix (), it is established that
the components of diverge as with
where is the set of frequencies of the
passband/stopband crossovers at the edges of the first Brillouin zone. The
divergence disappears for a homogeneous medium. Mathematical and physical
aspects of this observation are discussed. Explicit analytical examples of
and of its diverging asymptotics at
are provided for a model of scalar waves in a two-component periodic structure.
The case of high contrast due to stiff/soft layers or soft springs is
elaborated. Special attention in this case is given to the asymptotics of
near the first stopband that occurs at the Brillouin-zone
edge at arbitrary low frequency. The link to the quasi-static asymptotics of
the same near the point is also elucidated.Comment: 20 pages, 1 figur
Using SDO/HMI magnetograms as a source of the solar mean magnetic field data
The solar mean magnetic field (SMMF) provided by the Wilcox Solar Observatory
(WSO) is compared with the SMMF acquired by the \textit{Helioseismic and
Magnetic Imager} (HMI) onboard the \textit{Solar Dynamic Observatory} (SDO). We
found that despite the different spectral lines and measurement techniques used
in both instruments the Pearson correlation coefficient between these two
datasets equals 0.86 while the conversion factor is very close to unity: B(HMI)
= 0.99(2)B(WSO). We also discuss artifacts of the SDO/HMI magnetic field
measurements, namely the 12 and 24-hour oscillations in SMMF and in sunspots
magnetic fields that might be caused by orbital motions of the spacecraft. The
artificial harmonics of SMMF reveal significant changes in amplitude and the
nearly stable phase. The connection between the 24-hour harmonic amplitude of
SMMF and the presence of sunspots is examined. We also found that opposite
phase artificial 12 and/or 24-hour oscillations exist in sunspots of opposite
polarities.Comment: 13 pages, 9 figure
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