Pole Decomposition Research Articles

Comparison of frequency domain measures based on spectral decomposition for spontaneous baroreflex sensitivity assessment after Acute Myocardial Infarction

The objective of this study is to present a new method to assess in the frequency domain the directed interactions between the spontaneous variability of systolic arterial pressure (SAP) and heart period (HP) from their linear model representation, and to apply it for studying the baroreflex control of arterial pressure in healthy physiological states and after acute myocardial infarction (AMI). The method is based on pole decomposition of the model transfer function and on the following evaluation of causal measures of coupling and gain from the poles associated to low frequency (0.04−0.15 Hz) oscillatory components. It is compared with traditional non-causal approaches for the spectral analysis of the baroreflex gain, and with causal approaches based on the directed coherence, in a group of AMI patients and in Young and Old healthy controls studied at rest and during head-up tilt. Analysis of feedforward interactions from RR to SAP is also performed. Our results support the importance of using local causal approaches to quantify separately baroreflex and feedforward interactions between RR and SAP, allowing both to confirm known physiological behaviors (e.g., weaker baroreflex effectiveness and lower sensitivity after AMI) and to suggest novel findings (e.g., preserved low frequency baroreflex response to postural stress after AMI). We also find that the postural stress alters feedforward interactions selectively across groups, being related to decreased coupling only in Young and to increased gain mostly in AMI. These results have a significance for the clinical assessment of the baroreflex and the physiological evaluation of cardiovascular interactions.

Reflection identities of harmonic sums and pole decomposition of BFKL eigenvalue

We analyze the recent results of next-to-next-to-leading (NNLO) singlet BFKL eigenvalue in [Formula: see text] SYM written in terms of harmonic sums. The nested harmonic sums building known NNLO BFKL eigenvalue for specific values of the conformal spin have poles at negative integers. We sort the harmonic sums according to the complexity with respect to their weight and depth and use their pole decomposition in terms of the reflection identities to find the most complicated terms of NNLO BFKL eigenvalue for an arbitrary value of the conformal spin. The obtained result is compatible with the Bethe–Salpeter approach to the BFKL evolution.

The analytic structure of the BFKL equation and reflection identities of harmonic sums at weight five

We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N[Formula: see text]=[Formula: see text]4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N[Formula: see text]=4[Formula: see text]SYM, i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

Reflection Identities of Harmonic Sums of Weight Four

In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

Bound states in the continuum and time evolution of the generalized eigenfunctions

We study the Jost solutions for the scattering problem of a von Neumann-Wigner type potential, constructed by means of a two times iterated and completely degenerated Darboux transformation. We show that for a particular energy the unnormalizedJost solutions coalesce to give rise to a Jordan cycle of rank two. Performing a pole decomposition of the normalized Jost solutions we find the generalized eigenfunctions: one is a normalizable function corresponding to the bound state in the continuum and the other is a bounded, non-normalizable function. We obtain the time evolution of these functions as pseudo-unitary, characteristic of a pseudo-Hermitian system.

Stress-dependent normal-mode frequencies from the effective mass of granular matter.

A zero-temperature critical point has been invoked to control the anomalous behavior of granular matter as it approaches jamming or mechanical arrest. Criticality manifests itself in an anomalous spectrum of low-frequency normal modes and scaling behavior near the jamming transition. The critical point may explain the peculiar mechanical properties of dissimilar systems such as glasses and granular materials. Here we study the critical scenario via an experimental measurement of the normal modes frequencies of granular matter under stress from a pole decomposition analysis of the effective mass. We extract a complex-valued characteristic frequency which displays scaling |ω (σ)| ∼ σΩ' with vanishing stress σ for a variety of granular systems. The critical exponent is smaller than that predicted by mean-field theory opening new challenges to explain the exponent for frictional and dissipative granular matter. Our results shed light on the anomalous behavior of stress-dependent acoustics and attenuation in granular materials near the jamming transition.

Integrand reduction for two-loop scattering amplitudes through multivariate polynomial division

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Gr\"obner basis, we can derive a simple integrand recurrence relation that generates the multiparticle pole decomposition for integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$ supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the integrand-reduction method to all orders in perturbation theory.

Scattering amplitudes from multivariate polynomial division

We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the weak Nullstellensatz theorem and on the division modulo the Gröbner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrized by a number of coefficients equal to the number of solutions of the cut itself.

Premixed Flame Propagation in Channels of Varying Width

A theory of flame propagation in two-dimensional channels with adiabatic and impermeable curved walls is developed within the framework of the on-shell description of premixed flames. Employing the Green function appropriate to the given channel geometry, an implicit integral representation for the burnt gas velocity is constructed. It is next used to derive an explicit expression for the rotational component of the gas velocity near the flame front by successive separation of irrotational contributions. We prove that this separation can be performed in a way consistent with boundary conditions at the channel walls. As a result, the unknown irrotational component can be projected out by applying a dispersion relation stemming from its analyticity, thus leading to a closed system of equations for the on-shell fresh gas velocity and the flame front position. These equations show that, in addition to the usual nonlocality associated with potential flows, vorticity produced by a curved flame leads to specific nonlocal spatial and temporal influence of the channel geometry on the flame evolution. To elucidate this influence, three special cases are considered in more detail: a steady flame stabilized by incoming flow in a bottle-shaped channel, quasi-steady flames, and unsteady flames with small gas expansion propagating in channels of slowly varying width. In the last case, analytical solutions of the equations derived for the front shape are obtained in the first post-Sivashinsky approximation on using the method of pole decomposition.

A Transient Boundary Element Method for Acoustic Scattering from Mixed Regular and Thin Rigid Bodies

Boundary Element Methods (BEMs) may be used to predict the scattering of sound by obstacles, which has accelerated the prototyping of new room acoustic treatments such as diffusers. Unlike the more popular frequency domain method, the time domain BEM is usually solved in an iterative manner which means it can exhibit instability, a crucial impediment to its widespread use. These instabilities are primarily associated with the resonance of cavities formed by closed surface sections, but may also be caused by discretisation or integration error corrupting physical damped resonances. Regular BEM implementations cannot model objects with thin sections due to a phenomenon known as Thin Shape Breakdown. This paper develops an algorithm which combines an accepted approach for modelling thin plates with the Combined Field Integral Equation which eradicates cavity resonances, thereby permitting models of mixed regular and thin bodies. Accuracy and stability are tested by comparison to verified frequency domain BEMs, examination of the transient response, and pole decomposition. This is done for a simple obstacle and a Schroeder diffuser, which comprises a series of wells separated by thin fins. The approach is successful but universal stability cannot be guaranteed for the diffuser. It is suggested that instability is caused by the lightly damped resonances of the wells being corrupted into divergent behaviour by numerical errors.

Extension of the pole decomposition for the multidimensional Burgers equation

It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity nu and no force are of the form u=-2nu inverted Delta ln P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on time. Such solutions have pole singularities on the complex algebraic varieties.

Fine structure of the low-frequency spectra of heart rate and blood pressure.

BackgroundThe aim of this study was to explore the principal frequency components of the heart rate and blood pressure variability in the low frequency (LF) and very low frequency (VLF) band. The spectral composition of the R–R interval (RRI) and systolic arterial blood pressure (SAP) in the frequency range below 0.15 Hz were carefully analyzed using three different spectral methods: Fast Fourier transform (FFT), Wigner-Ville distribution (WVD), and autoregression (AR). All spectral methods were used to create time–frequency plots to uncover the principal spectral components that are least dependent on time. The accurate frequencies of these components were calculated from the pole decomposition of the AR spectral density after determining the optimal model order – the most crucial factor when using this method – with the help of FFT and WVD methods.ResultsSpectral analysis of the RRI and SAP of 12 healthy subjects revealed that there are always at least three spectral components below 0.15 Hz. The three principal frequency components are 0.026 ± 0.003 (mean ± SD) Hz, 0.076 ± 0.012 Hz, and 0.117 ± 0.016 Hz. These principal components vary only slightly over time. FFT-based coherence and phase-function analysis suggests that the second and third components are related to the baroreflex control of blood pressure, since the phase difference between SAP and RRI was negative and almost constant, whereas the origin of the first component is different since no clear SAP–RRI phase relationship was found.ConclusionThe above data indicate that spontaneous fluctuations in heart rate and blood pressure within the standard low-frequency range of 0.04–0.15 Hz typically occur at two frequency components rather than only at one as widely believed, and these components are not harmonically related. This new observation in humans can help explain divergent results in the literature concerning spontaneous low-frequency oscillations. It also raises methodological and computational questions regarding the usability and validity of the low-frequency spectral band when estimating sympathetic activity and baroreflex gain.

Comparison of experiments and a nonlinear model equation for spatially developing flame instability

We extend the Michelson–Sivashinsky equation, for arbitrary gas density ratio, to model the dynamics of an inclined anchored flame front. A flow velocity component tangential to the flame front transforms temporally developing structures into spatially developing ones. Closer investigation shows that a transition from absolute to convective instability should occur for a finite flow velocity. Using pole decomposition, we give a particular solution to this equation in the limit of high flow velocity compared to laminar flame speed. The analytical results are compared to measurements on a two-dimensional laminar flame. The growth rate, shape, and amplitude of the saturated structures, as well as the width of the crossover region are all well described. However, we experimentally observe the presence of a large scale transverse gas circulation not predicted by the analysis. We attribute this effect to the streamwise variation of the pressure jump across the flame brush.

Fast root MUSIC algorithm

The capabilities of the pole decomposition multiple signal classification (root-MUSIC) algorithm have been enhanced by the development of a simpler technique that allows the algorithm to operate at a much lower computational cost than was previously possible. The authors develop the approach into a new algorithm pertinent to the uniform linear array (ULA). Simulations are presented and the results evaluated against classical root-MUSIC.

Geometry of developing flame fronts: Analysis with pole decomposition.

The roughening of expanding flame fronts by the accretion of cusp-like singularities is a fascinating example of the interplay between instability, noise and nonlinear dynamics that is reminiscent of self-fractalization in Laplacian growth patterns. The nonlinear integro-differential equation that describes the dynamics of expanding flame fronts is amenable to analytic investigations using pole decomposition. This powerful technique allows the development of a satisfactory understanding of the qualitative and some quantitative aspects of the complex geometry that develops in expanding flame fronts.

DOA estimation by ARMA modelling and pole decomposition

A high-resolution DOA-estimation technique is proposed to deal with unknown noise-spatial-covariance structure and unknown array-sensor gain. By modelling the source signals as autoregressive moving-average (ARMA) processes with unknown parameters, a formula is derived which relates the source DOAs with the source poles and array-covariance functions. A virtual data matrix is formed, independent of the sensor-gain uncertainty and noise covariance, and a factorisation of this virtual data matrix shows that the subspace-based techniques can be directly applied to estimate the source DOAs. This technique has the advantage that it requires neither the prior knowledge about the sensor-noise covariance nor the sensor-gain calibration. Simulation results are presented to show the effectiveness of the technique and comparisons with the MUSIC algorithm are also included.

Analytical Solution of the Equation Describing the Outward Propagating Wrinkled flame

ABSTRACT The Sivashinsky integral equation, modelling the hydrodynamic instability of the outward propagating flame has exact solutions that can be derived by a modified method of pole decomposition.

Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts

Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts

Pole-zero modeling of speech based on high-order pole model fitting and decomposition method

In this paper four pole-zero modeling algorithms of clean and noisy speech have been studied in a unified approach that is based on high-order pole model fitting and decomposition method. They are autocorrelation prediction (AP), modified Yule-Walker (MYW), modified least square (MLS), and modified least square with autocorrelation compensation (MLSAC) methods. They involve only linear equations, and therefore are computationally efficient. Among these algorithms, the MLSAC method appears to be the most effective in spectral envelope estimation of noisy as well as clean speech. According to our simulation results, the improvement resulting from the use of the MLSAC pole-zero model for noisy speech is equivalent to increasing signal-to-noise ratio (SNR) by about 5 dB when SNR of input speech is 10 dB or less. The use of a pole-zero model in multirate vocoding is also discussed.

Estimation of time differences of arrival by pole decomposition

Time differences of arrival (TDOA's) of emitter wave fronts to a spatially distributed array of sensors can be used to determine the source location. In this paper, we suggest a new method of TDOA estimation for multiple unknown autoregressive moving average (ARMA) sources and additive noise that may be correlated between the sensors. We derive a theoretical formula that only uses the receiver cross spectra and the source poles for the TDOA determination. The poles are estimated by a least squares technique, and two methods are suggested for the estimation of the cross spectra which allow tradeoffs between computational complexity and accuracy. A new time delay model is derived and used to show the applicability of the methods for noninteger TDOA's. Results from simulations illustrate the performance of the algorithm by Monte Carlo trials and compare it to the Cramer-Rao bound.