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Structural equation modeling. --- Analysis of covariance. --- Linear models (Statistics) --- Multilevel models (Statistics)

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Neuroimaging post-stroke has the potential to uncover underlying principles of disturbed hand function and recovery characterizing defined patient groups, including their long term course as well as individual variations. The methods comprise functional magnetic resonance imaging (MRI) measuring task related activation as well as resting state. Functional MRI may be complemented by arterial spin labeling (ASL) MRI to investigate slowly varying blood flow and associated changes in brain function. For structural MRI robust and accurate computational anatomical methods like voxel-based morphometry and surface based techniques are available. The investigation of the connectivity among brain regions and disruption after stroke is facilitated by diffusion tensor imaging (DTI). Intra- and interhemispheric coherence may be studied by electromagnetic techniques such as electroencephalography and transcranial magnetic stimulation. Consecutive phases of stroke recovery (acute, subacute, early chronic and late chronic stages) are each distinguished by intrinsic processes. The site and size of lesions entail partially different functional implications. New strategies to establish functional specificity of a lesion site include calculating contrast images between patients exhibiting a specific disorder and control subjects without the disorder. Large-size lesions often imply poor cerebral blood flow which impedes recovery significantly and possibly interferes with BOLD response of functional MRI. Thus, depending on the site and size of the infarct lesion the patterns of recovery will vary. These include recovery sensu stricto in the perilesional area, intrinsic compensatory mechanisms using alternative cortical and subcortical pathways, or behavioral compensatory strategies e.g. by using the non-affected limb. In this context, behavioral and neuroimaging measures should be developed and employed to delineate aspects of learning during recovery. Of special interest in recovery of hand paresis is the interplay between sensory and motor areas in the posterior parietal cortex involved during reaching and fine motor skills as well as the interaction with the contralesional hemisphere. The dominant disability should be characterized, from the level of elementary to hierarchically higher processes such as neglect, apraxia and motor planning. In summary, this Research Topic covers new trends in state of the art neuroimaging of stroke during recovery from upper limb paresis. Integration of behavioral and neuroimaging findings in probabilistic brain atlases will further advance knowledge about stroke recovery.

stroke recovery --- Motor Imagery --- structural covariance --- Somatosensory Disorders --- perilesional plasticity --- network reorganization --- multimodal neuroimaging --- Neurorehabilitation --- computational biophysical modeling --- motor control

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This book is about the coordinate-free, or geometric, approach to the theory of linear models; more precisely, Model I ANOVA and linear regression models with non-random predictors in a finite-dimensional setting. This approach is more insightful, more elegant, more direct, and simpler than the more common matrix approach to linear regression, analysis of variance, and analysis of covariance models in statistics. The book discusses the intuition behind and optimal properties of various methods of estimating and testing hypotheses about unknown parameters in the models. Topics covered range from linear algebra, such as inner product spaces, orthogonal projections, book orthogonal spaces, Tjur experimental designs, basic distribution theory, the geometric version of the Gauss-Markov theorem, optimal and non-optimal properties of Gauss-Markov, Bayes, and shrinkage estimators under assumption of normality, the optimal properties of F-test, and the analysis of covariance and missing observations.

Linear models (Statistics) --- Analysis of variance --- Regression Analysis --- Analysis of covariance --- Analysis of variance. --- Regression analysis. --- Analysis of covariance. --- Covariance analysis --- Regression analysis --- Analysis, Regression --- Linear regression --- Regression modeling --- Multivariate analysis --- Structural equation modeling --- ANOVA (Analysis of variance) --- Variance analysis --- Mathematical statistics --- Experimental design --- Models, Linear (Statistics) --- Mathematical models --- Statistics --- Mathematical Sciences --- Probability

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This book primarily addresses the optimality aspects of covariate designs. A covariate model is a combination of ANOVA and regression models. Optimal estimation of the parameters of the model using a suitable choice of designs is of great importance; as such choices allow experimenters to extract maximum information for the unknown model parameters. The main emphasis of this monograph is to start with an assumed covariate model in combination with some standard ANOVA set-ups such as CRD, RBD, BIBD, GDD, BTIBD, BPEBD, cross-over, multi-factor, split-plot and strip-plot designs, treatment control designs, etc. and discuss the nature and availability of optimal covariate designs. In some situations, optimal estimations of both ANOVA and the regression parameters are provided. Global optimality and D-optimality criteria are mainly used in selecting the design. The standard optimality results of both discrete and continuous set-ups have been adapted, and several novel combinatorial techniques have been applied for the construction of optimum designs using Hadamard matrices, the Kronecker product, Rao-Khatri product, mixed orthogonal arrays to name a few.

Statistics. --- Statistics and Computing/Statistics Programs. --- Statistics for Business/Economics/Mathematical Finance/Insurance. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Statistics for Life Sciences, Medicine, Health Sciences. --- Mathematical statistics. --- Economics --- Statistique --- Statistique mathématique --- Analysis of covariance. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Covariance analysis --- Regression analysis --- Statistics for Business, Management, Economics, Finance, Insurance. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Statistics .

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The central object of the book is a subtle scalar Riemannian curvature quantity in even dimensions which is called Branson’s Q-curvature. It was introduced by Thomas Branson about 15 years ago in connection with an attempt to systematise the structure of conformal anomalies of determinants of conformally covariant differential operators on Riemannian manifolds. Since then, numerous relations of Q-curvature to other subjects have been discovered, and the comprehension of its geometric significance in four dimensions was substantially enhanced through the studies of higher analogues of the Yamabe problem. The book attempts to reveal some of the structural properties of Q-curvature in general dimensions. This is achieved by the development of a new framework for such studies. One of the main properties of Q-curvature is that its transformation law under conformal changes of the metric is governed by a remarkable linear differential operator: a conformally covariant higher order generalization of the conformal Laplacian. In the new approach, these operators and the associated Q-curvatures are regarded as derived quantities of certain conformally covariant families of differential operators which are naturally associated to hypersurfaces in Riemannian manifolds. This method is at the cutting edge of several central developments in conformal differential geometry in the last two decades such as Fefferman-Graham ambient metrics, spectral theory on Poincaré-Einstein spaces, tractor calculus, and Cartan geometry. In addition, the present theory is strongly inspired by the realization of the idea of holography in the AdS/CFT-duality. This motivates the term holographic descriptions of Q-curvature.

Analysis of covariance. --- Curvature. --- Differential operators. --- Geometry, Riemannian. --- Geometry, Riemannian --- Curvature --- Differential operators --- Analysis of covariance --- Mathematics --- Geometry --- Physical Sciences & Mathematics --- Riemann geometry --- Riemannian geometry --- Covariance analysis --- Operators, Differential --- Mathematics. --- Topological groups. --- Lie groups. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Differential geometry. --- Physics. --- Differential Geometry. --- Topological Groups, Lie Groups. --- Global Analysis and Analysis on Manifolds. --- Mathematical Methods in Physics. --- Regression analysis --- Differential equations --- Operator theory --- Calculus --- Curves --- Surfaces --- Generalized spaces --- Geometry, Non-Euclidean --- Semi-Riemannian geometry --- Global differential geometry. --- Topological Groups. --- Global analysis. --- Mathematical physics. --- Physical mathematics --- Physics --- Groups, Topological --- Continuous groups --- Geometry, Differential --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Differential geometry --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Topology