Group Analysis Method Research Articles

Steady-state oscillations in continuously inhomogeneous medium described by the generalized darboux equation

We refine a result due to L. V. Ovsyannikov on the general formof the second order linear differential equations with a nonzero generalized Laplace which are invariant admitting a Lie group of transformations of the maximal order with n > 2 independent variables for which the associated Riemannian spaces have nonzero curvature. We show that the set of these equations is exhausted by the generalized Darboux equation and the Ovsyannikov equation. We find the operators acting on the set of solutions in every one-parameter family of generalized Darboux equations. For the elliptic generalized Darboux equation possessing the maximal symmetry and describing steadystate oscillations in continuously inhomogeneous medium with a degeneration hyperplane, the group analysis methods yield the exact solutions to boundary value problems for some regions (a generalized Poisson formula) which in particular can be the test solutions in simulating steadystate oscillations in continuously inhomogeneous media.

Role of Linear Elasticity in Pile Group Analysis and Load Test Interpretation

This paper compares linear-elastic and nonlinear pile group analysis methods through settlement analyses of hypothetical scenarios and real case studies, and elaborates on the implications for interpretation of pile load test data. Comparisons between linear-elastic and nonlinear methods justify the proposition that pile-to-pile interaction is dominated by linear elasticity, characterized by the small-strain soil stiffness. As the size of a pile group increases, nonlinearity in individual pile behavior becomes overwhelmed by the interaction effects. In such cases, similar estimates will be achieved by both linear and nonlinear methods if the soil modulus is derived from the initial tangent, rather than some secant stiffness, assessed from the load test data. The study clarifies the capabilities and limitations of linear elasticity in pile group analysis and provides guidance on pile test interpretation for analysis of pile group response.

Invariant solutions of integro-differential Vlasov–Maxwell equations in Lagrangian variables by Lie group analysis

The Lie point symmetries of the Vlasov–Maxwell system in Lagrangian variables are investigated by using a direct method for symmetry group analysis of integro-differential equations, with emphasis on solving nonlocal determining equations. All similarity reduction forms for the system are obtained by using different approaches and some analytical and numerical solutions are presented.

Petrophysics laws as invariants of filtration model

In this paper the methods of Lie group analysis are used to investigate symmetry properties of differential equations that describe filtration of a two-phase liquid in a porousmedia. Lie algebra of operators of a group of equivalence transformations is calculated. Invariants with respect to a subgroup of the group of equivalence transformations are used for constructing the functional dependencies which define arbitrary parameters of the model. It is shown that one of the invariants of a subalgebra of operators of equivalence transformations is the Timur’s law, describing a relation between absolute permeability, porosity and residual water saturation.

Discovering structure in the space of fMRI selectivity profiles

We present a method for discovering patterns of selectivity in fMRI data for experiments with multiple stimuli/tasks. We introduce a representation of the data as profiles of selectivity using linear regression estimates, and employ mixture model density estimation to identify functional systems with distinct types of selectivity. The method characterizes these systems by their selectivity patterns and spatial maps, both estimated simultaneously via the EM algorithm. We demonstrate a corresponding method for group analysis that avoids the need for spatial correspondence among subjects. Consistency of the selectivity profiles across subjects provides a way to assess the validity of the discovered systems. We validate this model in the context of category selectivity in visual cortex, demonstrating good agreement with the findings based on prior hypothesis-driven methods.

Symmetries of shallow water equations on a rotating plane

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.

Canonical Correlation Analysis for Data Fusion and Group Inferences

We have presented two CCA-based approaches for data fusion and group analysis of biomedical imaging data and demonstrated their utility on fMRI, sMRI, and EEG data. The results show that CCA and M-CCA are powerful tools that naturally allow the analysis of multiple data sets. The data fusion and group analysis methods presented are completely data driven, and use simple linear mixing models to decompose the data into their latent components. Since CCA and M-CCA are based on second-order statistics they provide a relatively lessstrained solution as compared to methods based on higherorder statistics such as ICA. While this can be advantageous, the flexibility also tends to lead to solutions that are less sparse than those obtained using assumptions of non-Gaussianity-in particular superGaussianity-at times making the results more difficult to interpret. Thus, it is important to note that both approaches provide complementary perspectives, and hence it is beneficial to study the data using different analysis techniques.

Stability of bifurcating solutions of the problem about capillary‐gravity surface waves in spatial layer of floating fluid

AbstractIn prolongation of our previous investigations on capillary‐gravity surface waves in spatial fluid layers the stability of the bifurcating families of solutions in the horizontal layers of the floating (and without flotation) incompressible heavy capillary fluid is considered. The assumption about layer depth simplifies the proof of the existence of bifurcating solutions at the high dimensions of the linearized operator degeneracy, computation of their asymptotics and as the main subject of this communication the investigation of their stability, relative to perturbations with the same symmetry as bifurcating solutions. Group analysis methods of differential equations are used. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Painlevé analysis, Lie symmetries, and exact solutions for the time-dependent coefficients Gardner equations

In this paper, the three variable-coefficient Gardner (vc-Gardner) equations are considered. By using the Painleve analysis and Lie group analysis method, the Painleve properties and symmetries for the equations are obtained. Then the exact solutions generated from the symmetries and Painleve analysis are presented.

The dependence on noise and inter-subject variability of different group analysis methods in dynamic causal modeling

The dependence on noise and inter-subject variability of different group analysis methods in dynamic causal modeling

Optical Solitons with Parabolic and Dual-Power Law Nonlinearity via Lie Symmetry Analysis

This paper studies optical solitons by the method of Lie group analysis. The stationary 1-soliton solution is obtained for the nonlinear Schrödinger's equation with parabolic and dual-power law nonlinearity.

MicroRNA Set: A Novel Way to Uncover the Potential Black Box of Chronic Heart Failure in MicroRNA Microarray Analysis

As the prime criminal among all the cardio-vascular diseases, chronic heart failure (CHF) is still far from being fully understood after decades of study by researchers. The booming bioinformatics studies, especially the microRNA (miRNA) microarray analysis, have significantly accelerated the uncovering of underlying mechanisms of human diseases. However, these miRNA researches mainly focus on single miRNA, paying less attention to the group characteristics of miRNAs, which may ignore the group characteristics of miRNAs. Here we introduce a novel miRNA set concept incorporation with a group analysis method of CHF miRNA microarray expression. Our results show great accordance with previous studies, and also reveal potential characteristics of miRNAs in CHF. Furthermore, this novel miRNA set approach may give us new insights into other diseases studies as well.

Local sphere-based co-registration for SAM group analysis in subjects without individual MRI

Synthetic aperture magnetometry (SAM) is a powerful MEG source localization method to analyze evoked as well as induced brain activity. To gain structural information of the underlying sources, especially in group studies, individual magnetic resonance images (MRI) are required for co-registration. During the last few years, the relevance of MEG measurements on understanding the pathophysiology of different diseases has noticeable increased. Unfortunately, especially in patients and small children, structural MRI scans cannot always be performed. Therefore, we developed a new method for group analysis of SAM results without requiring structural MRI data that derives its geometrical information from the individual volume conductor model constructed for the SAM analysis. The normalization procedure is fast, easy to implement and integrates seamlessly into an existing landmark based MEG-MRI co-registration procedure. This new method was evaluated on different simulated points as well as on a pneumatic index finger stimulation paradigm analyzed with SAM. Compared with an established MRI-based normalization procedure (SPM2) the new method shows only minor errors in single subject results as well as in group analysis. The mean difference between the two methods was about 4 mm for the simulated as well as for finger stimulation data. The variation between individual subjects was generally higher than the error induced by the missing MRIs. The method presented here is therefore sufficient for most MEG group studies. It allows accomplishing MEG studies with subject groups where MRI measurements cannot be performed.

Group analysis method for fission track thermochronology

A methodology to obtain ages and thermal histories of sets of apatite samples from localities with geologically compatible characteristics is described. A methodology exploring the fact that samples with similar geological characteristics should present the same thermal history is proposed. This approach can contribute for the obtainment of more conclusive results by analysing fewer samples than it is necessary when the samples are individually analysed. In order to determine the ages, we use the absolute neutron dosimetry through thin films of natural uranium along with λ f = 8.46 × 10 - 17 a - 1 . As an example of application of the proposed methodology, we analyse samples collected in a Brazilian region, São Francisco Craton, which experienced low tectonic activity.

New method to determine the total carbonyl functional group content in extractable particulate organic matter by tandem mass spectrometry

A functional group analysis method was developed to determine the quantitative content of carbonyl functional groups in atmospheric particulate organic matter (POM) using constant neutral loss scanning-tandem mass spectrometry (CNLS-MS/MS). The neutral loss method consists in monitoring the loss of a neutral fragment produced by the fragmentation of a precursor ion in a collision cell. The only ions detected are the daughter ions resulting from the loss of the neutral fragment under study. Then, scanning the loss of a neutral fragment characteristic of a functional group enables the selective detection of the compounds bearing the chemical function under study within a complex mixture. The selective detection of carbonyl functional groups was achieved after derivatization with pentafluorophenylhydrazine (PFPH) by monitoring the neutral loss of C(6)F(5)N (181 amu), which was characteristic of a large panel of derivatized carbonyl compounds. The method was tested on 25 reference mixtures of different composition, all containing 24 carbonyl compounds at randomly determined concentrations. The repeatability and calibration tests were satisfying as they resulted in a relative standard deviation below 5% and a linear range between 0.01 and 0.65 mM with a calculated detection limit of 0.0035 mM. Also, the relative deviation induced by changing the composition of the mixture while keeping the total concentration of carbonyl functional groups constant was less than 20%. These reliability experiments demonstrate the high robustness of the developed procedure for accurate carbonyl functional group measurement, which was applied to atmospheric POM samples.

Diffusion in flows with a linearly nonuniform velocity profile

Methods of group analysis are applied to determine the Green’s function in the infinite space of the test particle diffusion equation for flows with a linearly nonuniform velocity profile. Using this function, the solutions of the Cauchy problem are obtained for certain particular initial conditions.

Support vector machine learning-based fMRI data group analysis

To explore the multivariate nature of fMRI data and to consider the inter-subject brain response discrepancies, a multivariate and brain response model-free method is fundamentally required. Two such methods are presented in this paper by integrating a machine learning algorithm, the support vector machine (SVM), and the random effect model. Without any brain response modeling, SVM was used to extract a whole brain spatial discriminance map (SDM), representing the brain response difference between the contrasted experimental conditions. Population inference was then obtained through the random effect analysis (RFX) or permutation testing (PMU) on the individual subjects' SDMs. Applied to arterial spin labeling (ASL) perfusion fMRI data, SDM RFX yielded lower false-positive rates in the null hypothesis test and higher detection sensitivity for synthetic activations with varying cluster size and activation strengths, compared to the univariate general linear model (GLM)-based RFX. For a sensory–motor ASL fMRI study, both SDM RFX and SDM PMU yielded similar activation patterns to GLM RFX and GLM PMU, respectively, but with higher t values and cluster extensions at the same significance level. Capitalizing on the absence of temporal noise correlation in ASL data, this study also incorporated PMU in the individual-level GLM and SVM analyses accompanied by group-level analysis through RFX or group-level PMU. Providing inferences on the probability of being activated or deactivated at each voxel, these individual-level PMU-based group analysis methods can be used to threshold the analysis results of GLM RFX, SDM RFX or SDM PMU.

Parietal and superior frontal visuospatial maps activated by pointing and saccades

A recent study from our laboratory demonstrated that parietal cortex contains a map of visual space related to saccades and spatial attention and identified this area as the likely human homologue of the lateral intraparietal (LIP). A human homologue for the parietal reach region (PRR), thought to preferentially encode planned hand movements, has also been recently proposed. Both of these areas, originally identified in the macaque monkey, have been shown to encode space with eye-centered coordinates. Functional magnetic resonance imaging (fMRI) of humans was used to test the hypothesis that the putative human PRR contains a retinotopic map recruited by finger pointing but not saccades and to test more generally for differences in the visuospatial maps recruited by pointing and saccades. We identified multiple maps in both posterior parietal cortex and superior frontal cortex recruited for eye and hand movements, including maps not observed in previous mapping studies. Pointing and saccade maps were generally consistent within single subjects. We have developed new group analysis methods for phase-encoded data, which revealed subtle differences between pointing and saccades, including hemispheric asymmetries, but we did not find evidence of pointing-specific maps of visual space.

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.

Approximate Group Analysis and Multiple Time Scales Method for the Approximate Boussinesq Equation

This paper is devoted to investigation of the approximate Boussinesq equation by methods of the approximate symmetry analysis of partial differential equations with a small parameter developed by Baikov, Gazizov and Ibragimov. We combine these methods with the method of multiple time scales to extend the domain of definition of approximate group invariant solutions of the approximate Boussinesq equation.