40,495 research outputs found
Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems
Multidimensional matrix inversions provide a powerful tool for studying
multiple hypergeometric series. In order to extend this technique to elliptic
hypergeometric series, we present three new multidimensional matrix inversions.
As applications, we obtain a new elliptic Jackson summation, as well as
several quadratic, cubic and quartic summation formulas
Multilateral inversion of A_r, C_r and D_r basic hypergeometric series
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic
hypergeometric matrix inverse with applications to bilateral basic
hypergeometric series. This matrix inversion result was directly extracted from
an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and
involves two infinite matrices which are not lower-triangular. The present
paper features three different multivariable generalizations of the above
result. These are extracted from Gustafson's A_r and C_r extensions and of the
author's recent A_r extension of Bailey's 6-psi-6 summation formula. By
combining these new multidimensional matrix inverses with A_r and D_r
extensions of Jackson's 8-phi-7 summation theorem three balanced
very-well-poised 8-psi-8 summation theorems associated with the root systems
A_r and C_r are derived.Comment: 24 page
A new multivariable 6-psi-6 summation formula
By multidimensional matrix inversion, combined with an A_r extension of
Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7
summation is derived. By a polynomial argument this 8-phi-7 summation is
transformed to another multivariable 8-phi-7 summation which, by taking a
suitable limit, is reduced to a new multivariable extension of the
nonterminating 6-phi-5 summation. The latter is then extended, by analytic
continuation, to a new multivariable extension of Bailey's very-well-poised
6-psi-6 summation formula.Comment: 16 page
On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series
Using Krattenthaler's operator method, we give a new proof of Warnaar's
recent elliptic extension of Krattenthaler's matrix inversion. Further, using a
theta function identity closely related to Warnaar's inversion, we derive
summation and transformation formulas for elliptic hypergeometric series of
Karlsson-Minton-type. A special case yields a particular summation that was
used by Warnaar to derive quadratic, cubic and quartic transformations for
elliptic hypergeometric series. Starting from another theta function identity,
we derive yet different summation and transformation formulas for elliptic
hypergeometric series of Karlsson-Minton-type. These latter identities seem
quite unusual and appear to be new already in the trigonometric (i.e., p=0)
case.Comment: 16 page
Data-adaptive harmonic spectra and multilayer Stuart-Landau models
Harmonic decompositions of multivariate time series are considered for which
we adopt an integral operator approach with periodic semigroup kernels.
Spectral decomposition theorems are derived that cover the important cases of
two-time statistics drawn from a mixing invariant measure.
The corresponding eigenvalues can be grouped per Fourier frequency, and are
actually given, at each frequency, as the singular values of a cross-spectral
matrix depending on the data. These eigenvalues obey furthermore a variational
principle that allows us to define naturally a multidimensional power spectrum.
The eigenmodes, as far as they are concerned, exhibit a data-adaptive character
manifested in their phase which allows us in turn to define a multidimensional
phase spectrum.
The resulting data-adaptive harmonic (DAH) modes allow for reducing the
data-driven modeling effort to elemental models stacked per frequency, only
coupled at different frequencies by the same noise realization. In particular,
the DAH decomposition extracts time-dependent coefficients stacked by Fourier
frequency which can be efficiently modeled---provided the decay of temporal
correlations is sufficiently well-resolved---within a class of multilayer
stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators.
Applications to the Lorenz 96 model and to a stochastic heat equation driven
by a space-time white noise, are considered. In both cases, the DAH
decomposition allows for an extraction of spatio-temporal modes revealing key
features of the dynamics in the embedded phase space. The multilayer
Stuart-Landau models (MSLMs) are shown to successfully model the typical
patterns of the corresponding time-evolving fields, as well as their statistics
of occurrence.Comment: 26 pages, double columns; 15 figure
Tensor Analysis and Fusion of Multimodal Brain Images
Current high-throughput data acquisition technologies probe dynamical systems
with different imaging modalities, generating massive data sets at different
spatial and temporal resolutions posing challenging problems in multimodal data
fusion. A case in point is the attempt to parse out the brain structures and
networks that underpin human cognitive processes by analysis of different
neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the
multimodal, multi-scale nature of neuroimaging data is well reflected by a
multi-way (tensor) structure where the underlying processes can be summarized
by a relatively small number of components or "atoms". We introduce
Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network
notation in order to analyze these models. These diagrams not only clarify
matrix and tensor EEG and fMRI time/frequency analysis and inverse problems,
but also help understand multimodal fusion via Multiway Partial Least Squares
and Coupled Matrix-Tensor Factorization. We show here, for the first time, that
Granger causal analysis of brain networks is a tensor regression problem, thus
allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI
recordings shows the potential of the methods and suggests their use in other
scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE
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