2,236 research outputs found
Spectral Theory of Discrete Processes
We offer a spectral analysis for a class of transfer operators. These
transfer operators arise for a wide range of stochastic processes, ranging from
random walks on infinite graphs to the processes that govern signals and
recursive wavelet algorithms; even spectral theory for fractal measures. In
each case, there is an associated class of harmonic functions which we study.
And in addition, we study three questions in depth:
In specific applications, and for a specific stochastic process, how do we
realize the transfer operator as an operator in a suitable Hilbert space?
And how to spectral analyze once the right Hilbert space has
been selected? Finally we characterize the stochastic processes that are
governed by a single transfer operator.
In our applications, the particular stochastic process will live on an
infinite path-space which is realized in turn on a state space . In the case
of random walk on graphs , will be the set of vertices of . The
Hilbert space on which the transfer operator acts will then
be an space on , or a Hilbert space defined from an energy-quadratic
form.
This circle of problems is both interesting and non-trivial as it turns out
that may often be an unbounded linear operator in ; but even
if it is bounded, it is a non-normal operator, so its spectral theory is not
amenable to an analysis with the use of von Neumann's spectral theorem. While
we offer a number of applications, we believe that our spectral analysis will
have intrinsic interest for the theory of operators in Hilbert space.Comment: 34 pages with figures removed, for the full version with all the
figures please go to http://www.siue.edu/~msong/Research/spectrum.pd
Unconventional Approach with the Likelihood of Correlation Matrices
Numerical approximations are important research areas for dealing with complicated functional forms. Techniques for developing accurate and efficient calculation of combined likelihood functions in meta-analysis are studied. The first part of the thesis introduces a B-spline approximation for making a parsimonious model in the simplest case(2-dimensional case) of correlation structure. Inference about the correlation between vitamin C intake & vitamin C serum level is developed by using likelihood intervals and the MLE, along with comparison with conventional methods. The second part studies a multivariate numerical integration method for developing a better approximation of the likelihood for correlation matrices. Analyses for (1) intercorrelations among Math, Spatial and Verbal scores in an SAT exam and (2) intercorrelations among Cognitive Anxiety, Somatic Anxiety and Self Confidence from Competitive State Anxiety Inventory (CSAI-2) are explored. Algorithms to evaluate likelihood and to find the MLE is developed. Comparison with two conventional methods (joint asymptotic weighted average method & marginal asymptotic weighted average method) is shown
Structural and Correlation Effects in the Itinerant Insulating Antiferromagnetic Perovskite NaOsO3
The orthorhombic perovskite NaOsO3 undergoes a continuous metal-insulator
transition (MIT), accompanied by antiferromagnetic (AFM) order at T_N=410 K,
suggested to be an example of the rare Slater (itinerant) MIT. We study this
system using ab initio and related methods, focusing on the origin and nature
of magnetic ordering and the MIT. The rotation and tilting of OsO6 octahedra in
the GdFeO3 structure result in moderate narrowing the band width of the t_{2g}
manifold, but sufficient to induce flattening of bands and AFM order within the
local spin density approximation (LSDA), where it remains metallic but with a
deep pseudogap. Including on-site Coulomb repulsion U, at U_c ~2 eV a MIT
occurs only in the AFM state. Effects of spin-orbit coupling (SOC) on the band
structure seem minor as expected for a half-filled shell, but SOC
doubles the critical value U_c necessary to open a gap and also leads to large
magnetocrystalline energy differences in spite of normal orbital moments no
greater than 0.1. Our results are consistent with a Slater MIT driven by
magnetic order, induced by a combination of structurally-induced band narrowing
and moderate Coulomb repulsion, with SOC necessary for a full picture. Strong
p-d hybridization reduces the moment, and when bootstrapped by the reduced
Hund's rule coupling (proportional to the moment) gives a calculated moment of
~1 , consistent with the observed moment and only a third of the formal
value. We raise and discuss one important question: since this AFM
ordering is at q=0 (in the 20 atom cell) where nesting is a moot issue, what is
the microscopic driving force for ordering and the accompanying MIT?Comment: 9 page
Markov chains and generalized wavelet multiresolutions
We develop some new results for a general class of transfer operators, as they are used in a construction of multi-resolutions. We then proceed to give explicit and concrete applications. We further discuss the need for such a constructive harmonic analysis/dynamical systems approach to fractals
Infinite-Dimensional Measure Spaces and Frame Analysis
We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space â„‹, we study three cases of measures. We first show that, for â„‹ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain â„‹. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures
Reproducing Kernel Hilbert Space vs. Frame Estimates
We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn Hinto a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes
Filters and Matrix Factorization
We give a number of explicit matrix-algorithms for analysis/synthesis
in multi-phase filtering; i.e., the operation on discrete-time signals which
allow a separation into frequency-band components, one for each of the
ranges of bands, say N , starting with low-pass, and then corresponding
filtering in the other band-ranges. If there are N bands, the individual
filters will be combined into a single matrix action; so a representation of
the combined operation on all N bands by an N x N matrix, where the
corresponding matrix-entries are periodic functions; or their extensions to
functions of a complex variable. Hence our setting entails a fixed N x N
matrix over a prescribed algebra of functions of a complex variable. In the
case of polynomial filters, the factorizations will always be finite. A novelty
here is that we allow for a wide family of non-polynomial filter-banks.
Working modulo N in the time domain, our approach also allows for
a natural matrix-representation of both down-sampling and up-sampling.
The implementation encompasses the combined operation on input, filtering,
down-sampling, transmission, up-sampling, an action by dual filters,
and synthesis, merges into a single matrix operation. Hence our matrixfactorizations
break down the global filtering-process into elementary steps.
To accomplish this, we offer a number of adapted matrix factorizationalgorithms,
such that each factor in our product representation implements
in a succession of steps the filtering across pairs of frequency-bands; and so
it is of practical significance in implementing signal processing, including
filtering of digitized images. Our matrix-factorizations are especially useful
in the case of the processing a fixed, but large, number of bands
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