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Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres (n-spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres (n-spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.

Because of the advance in technologies, modern statistical studies often encounter linear models with the number of explanatory variables much larger than the sample size. Estimation and variable selection in these highdimensional problems with deterministic design points is very different from those in the case of random covariates, due to the identifiability of the highdimensional regression parameter vector. We show that a reasonable approach is to focus on the projection of the regression parameter vector onto the linear space generated by the design matrix. In this work, we consider the ridge regression estimator of the projection vector and propose to threshold the ridge regression estimator when the projection vector is sparse in the sense that many of its components are small. The proposed estimator has an explicit form and is easy to use in application. Asymptotic properties such as the consistency of variable selection and estimation and the convergence rate of the prediction mean squared error are established under some sparsity conditions on the projection vector. A simulation study is also conducted to examine the performance of the proposed estimator.

In this paper, we provide a projection-based analysis of the hh-version of the hybridizable discontinuous Galerkin methods for convection-diffusion equations on semimatching nonconforming meshes made of simplexes; the degrees of the piecewise polynomials are allowed to vary from element to element. We show that, for approximations of degree kk on all elements, the order of convergence of the error in the diffusive flux is k+1k+1 and that of a projection of the error in the scalar unknown is 11 for k=0k=0 and k+2k+2 for k>0k>0. We also show that, for the variable-degree case, the projection of the error in the scalar variable is hh times the projection of the error in the vector variable, provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes. These results hold for any (bounded) irregularity index of the nonconformity of the mesh. Moreover, our analysis can be extended to hypercubes.

We use spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.

An affine manifold X in the sense of differential geometry is a differentiable manifold admitting an atlas of charts with value in an affine space, with locally constant affine change of coordinates. Equivalently, it is a manifold whose tangent bundle admits a flat torsion free connection. Around 1955 Chern conjectured that the Euler characteristic of any compact affine manifold has to vanish. In this paper we prove Chern's conjecture in the case where X moreover admits a parallel volume form.

The biased-competition theory accounts for attentional effects at the single-neuron level: It predicts that the neuronal response to simultaneously-presented stimuli is a weighted average of the response to isolated stimuli, and that attention biases the weights in favor of the attended stimulus. Perception, however, relies not on single neurons but on larger neuronal populations. The responses of such populations are in part reflected in large-scale multivoxel fMRI activation patterns. Because the pooling of neuronal responses into blood-oxygen-level-dependent signals is nonlinear, fMRI effects of attention need not mirror those observed at the neuronal level. Thus, to bridge the gap between neuronal responses and human perception, it is fundamental to understand attentional influences in large-scale multivariate representations of simultaneously-presented objects. Here, we ask how responses to simultaneous stimuli are combined in multivoxel fMRI patterns, and how attention affects the paired response. Objects from four categories were presented singly, or in pairs such that each category was attended, unattended, or attention was divided between the two. In a multidimensional voxel space, the response to simultaneously-presented categories was well described as a weighted average. The weights were biased toward the preferred category in category-selective regions. Consistent with single-unit reports, attention shifted the weights by ≈30% in favor of the attended stimulus. These findings extend the biased-competition framework to the realm of large-scale multivoxel brain activations.

We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a \"physical condition\", related to the fact that the pressure of a fluid has to be positive.

This work presents the automated projection spectroscopy (APSY) method for the recording of discrete sets of j projections from N-dimensional (N ≥ 3) NMR experiments at operator-selected projection angles and automatic identification of the correlation cross peaks. The result from APSY is the fully automated generation of the complete or nearly complete peak list for the N-dimensional NMR spectrum from a geometric analysis of the j experimentally recorded, low-dimensional projections. In the present implementation of APSY, two-dimensional projections of the N-dimensional spectrum are recorded by using techniques developed for projection-reconstruction spectroscopy [Kupče, E. & Freeman, R. (2004) J. Am. Chem. Soc. 126, 6429-6440]. All projections are peak-picked with the available automated routine ATNOS. The previously undescribed algorithm GAPRO (geometric analysis of projections) uses vector algebra to identify subgroups of peaks in different projections that arise from the same resonance in the N-dimensional spectrum, and from these subgroups it calculates the peak positions in the N-dimensional frequency space. Unambiguous identification thus can be achieved for all cross peaks that are not overlapped with other peaks in at least one of the N dimensions. Because of the correlation between the positions of corresponding peaks in multiple projections, uncorrelated noise is efficiently suppressed, so that APSY should be quite widely applicable for correlation spectra of biological macromolecules, which have intrinsically low peak density in the N-dimensional spectral space.

The paper proposes one-to-one transformation of the vector of components $\\{{\\mathrm{Y}}_{\\mathrm{i}\\mathrm{n}}{\\}}_{\\mathrm{i}=1}^{\\mathrm{m}}$ of Pearson's chi-square statistic, ${\\mathrm{Y}}_{\\mathrm{i}\\mathrm{n}}=\\frac{{\\mathrm{\\nu }}_{\\mathrm{i}\\mathrm{n}}-\\mathrm{n}{\\mathrm{p}}_{\\mathrm{i}}}{\\sqrt{\\mathrm{n}{\\mathrm{p}}_{\\mathrm{i}}},\\mathrm{i}=1,\\mathrm{...},\\mathrm{m}$ , into another vector $\\{{\\mathrm{Z}}_{\\mathrm{i}\\mathrm{n}}{\\}}_{\\mathrm{i}=1}^{\\mathrm{m}}$ , which, therefore, contains the same \"statistical information,\" but is asymptotically distribution free. Hence any functional/test statistic based on $\\{{\\mathrm{Z}}_{\\mathrm{i}\\mathrm{n}}{\\}}_{\\mathrm{i}=1}^{\\mathrm{m}}$ is also asymptotically distribution free. Natural examples of such test statistics are traditional goodness-of-fit statistics from partial sums Σ I≤k Z in . The supplement shows how the approach works in the problem of independent interest: the goodness-of-fit testing of power-law distribution with the Zipf law and the Karlin—Rouault law as particular alternatives.

An extension to multivariate empirical mode decomposition (MEMD), termed adaptive-projection intrinsically transformed MEMD (APIT-MEMD), is proposed to cater for power imbalances and inter-channel correlations in real-world multichannel data. It is shown that the APIT-MEMD exhibits similar or better performance than MEMD for a large number of projection vectors, whereas it outperforms MEMD for the critical case of a small number of projection vectors within the sifting algorithm. We also employ the noise-assisted APIT-MEMD within our proposed intrinsic multiscale analysis framework and illustrate the advantages of such an approach in notoriously noise-dominated cooperative brain–computer interface (BCI) based on the steady-state visual evoked potentials and the P300 responses. Finally, we show that for a joint cognitive BCI task, the proposed intrinsic multiscale analysis framework improves system performance in terms of the information transfer rate.