Euclidean Mean Research Articles

Weakly Increasing Solutions of Equations with p-Mean Curvature Operator

Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When p≥2, the results highlight an analogy with an auxiliary equation with the p-Laplacian operator. The results are obtained using some comparison criteria for the principal solutions of a class of associated half-linear equations.

Mechanical mechanism of dip effect on bearing capacity of the pillar strength

Abstract The dip effect on pillar strength in underground ore-body mining is well established, but the variation in stress path (magnitude and direction of stress) due to changing inclination angles requires further study. Using elasticity theory, the Euclidean mean stress tensor characterizes the stress state in pillar zones. Numerical simulations provided the second-order tensor of peak stress for each pillar unit. Through tensor statistical analysis, the Euclidean mean stress tensor matrix was calculated, and its eigenvalues and eigenvectors, representing the magnitude and direction of the principal stress, were derived. This analysis explained the intrinsic dip effect on pillar strength through principal stress characteristics. Finally, the pillar strength envelope function for varying width-height ratios at any dip angle was obtained using the random gradient descent algorithm. Results indicate that in the peak stress state, the average principal stress directions of the pillar change with orebody dip angle, affecting the stress path. The average principal stress increases with pillar size due to increased constraints. These findings offer theoretical insights for pillar design and stability analysis.

Fetal ECG denoising using dynamic time warping template subtraction.

The analysis of the fetal electrocardiogram (ECG) requires to remove the mother ECG (mECG) from the abdominal ECG signals. Template subtraction is a method that consists in modeling and removing the mECG's mean period i.e. the signal waveform defined as the Euclidean mean of all periods. This mean period is then subtracted to all periods to extract the fetal ECG (fECG). Such a method is not accurate because each mECG's period is not correctly aligned with the mean period. We propose to take account of the diffeomorphism of each period to improve the precision of the model and remove the mECG more efficiently. The soft-dynamic time warping (DTW) algorithm is used to compute the mean mECG period and the alignment between the mean period and all periods. Our approach is compared to a classic template subtraction on synthetic and real databases. Results show that considering the dynamic time warping allows a better removal of the mECG. Clinical relevance - The template subtraction is modified in this paper to consider the time warping for each mother ECG's period in order to improve the fetal ECG extraction from the abdominal ECG.

Genetic diversity among cassava genotypes (Manihot esculenta crantz) collected in the south-central mesoregion of the state of Mato Grosso-Brazil

The aim of this research was to evaluate the genetic divergence between 164 genotypes of Manihot esculenta from the South-Central mesoregion of the State of Mato Grosso. The genotypes are from projects conducted by the Laboratory of Genetic Resources & Biotechnology of the University of the State of Mato Grosso, Cáceres-Mato Grosso (UNEMAT), and the Brazilian Public Agricultural Research Corporation - Agrosilvopastoral (EMBRAPA). The agronomic descriptors evaluated were plant height, height of first branching, branching levels, weight of the aerial part of the plant, total weight of the plant, number of roots per plant, average weight of roots per plant, yield of commercial roots, yield of non-commercial roots, number of rotten roots per plant and harvest index. For the analysis of genetic divergence, multivariate analysis based on the standardized Euclidean mean distance was employed, later performing the Hierarchical UPGMA and Tocher Optimization agglomerative methods. The degree of preservation of the genetic distances in the dendrogram was verified using the Cophenetic Correlation Coefficient. The Singh criterion was used to quantify the relative contribution of characteristics to genetic divergence. The genotypes presented genetic dissimilarity for the evaluated characteristics and based on the results of the dissimilarity matrix and groupings, it is recommended the crossings between the genotypes allocated in group II with the genotype allocated in group V, for the development of segregated populations with high genetic variability.

TractLearn: A geodesic learning framework for quantitative analysis of brain bundles

Deep learning-based convolutional neural networks have recently proved their efficiency in providing fast segmentation of major brain fascicles structures, based on diffusion-weighted imaging. The quantitative analysis of brain fascicles then relies on metrics either coming from the tractography process itself or from each voxel along the bundle.Statistical detection of abnormal voxels in the context of disease usually relies on univariate and multivariate statistics models, such as the General Linear Model (GLM). Yet in the case of high-dimensional low sample size data, the GLM often implies high standard deviation range in controls due to anatomical variability, despite the commonly used smoothing process. This can lead to difficulties to detect subtle quantitative alterations from a brain bundle at the voxel scale.Here we introduce TractLearn, a unified framework for brain fascicles quantitative analyses by using geodesic learning as a data-driven learning task. TractLearn allows a mapping between the image high-dimensional domain and the reduced latent space of brain fascicles using a Riemannian approach.We illustrate the robustness of this method on a healthy population with test-retest acquisition of multi-shell diffusion MRI data, demonstrating that it is possible to separately study the global effect due to different MRI sessions from the effect of local bundle alterations. We have then tested the efficiency of our algorithm on a sample of 5 age-matched subjects referred with mild traumatic brain injury.Our contributions are to propose:1/ A manifold approach to capture controls variability as standard reference instead of an atlas approach based on a Euclidean mean.2/ A tool to detect global variation of voxels’ quantitative values, which accounts for voxels’ interactions in a structure rather than analyzing voxels independently.3/ A ready-to-plug algorithm to highlight nonlinear variation of diffusion MRI metrics.With this regard, TractLearn is a ready-to-use algorithm for precision medicine.

Quantum divergences with p-power means

We study several properties of a family of quantum divergences with p-power means. We also show that the Hellinger distance with log Euclidean mean and the quantum Jensen-Shannon divergence can be rediscovered from our class of quantum distances via the limit processes.

Asymptotics for optimal controls for horizontal mean curvature flow

The solutions to surface evolution problems like mean curvature flow can be expressed as value functions of suitable stochastic control problems, obtained as limit of a family of regularised control problems. The control-theoretical approach is particularly suited for such problems for degenerate geometries like the Heisenberg group. In this situation a new type of singularities absent for the Euclidean mean curvature flow occurs, the so-called characteristic points. This paper investigates the asymptotic behaviour of the regularised optimal controls in the vicinity of such characteristic points.

Properties for the Fréchet mean in Billera-Holmes-Vogtmann treespace

The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Fréchet mean exists and is unique in treespace. This Fréchet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial algorithm for computing the Fréchet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Fréchet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Fréchet mean. This work generalizes to orthant spaces.

Decaying positive global solutions of second order difference equations with mean curvature operator

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is examined by proving new Sturm comparison theorems for linear difference equations and using a fixed point approach based on a linearization device. The process of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous cases are pointed out, too.

Can we estimate far-field stress using the mean of local stresses? An examination based on numerical simulations

In situ stress is an important parameter in rock mechanics, and reliable estimates of far-field stresses are indispensable for robust rock engineering analysis. Here, by using the combined finite-discrete element method, we simulate a series of stress fields of both synthetic and natural fracture networks to examine whether the Euclidean mean of local stresses can be used to estimate the far-field stress. The calculations show that given a large number of local stress measurements, their Euclidean mean gives a close approximation of the far-field stress state. Whereas when only a limited number of local stresses are available, the probability of obtaining a practically acceptable estimate of far-field stress increases when more stress measurements are involved. The required number of stress measurements for deriving an acceptable estimate varies with the geomechanical condition; in general, the larger the overall stress variability is, the more local stress measurements are needed. Our research findings suggest that given a limited number of stress measurements, which is often encountered in rock engineering projects, attention is needed when deriving the far-field stress state based on them, and simply using their mean as far-field stress for further rock structure design and numerical analysis may yield erroneous results.

Knowledge-Aided Structured Covariance Matrix Estimator Applied for Radar Sensor Signal Detection.

This study deals with the problem of covariance matrix estimation for radar sensor signal detection applications with insufficient secondary data in non-Gaussian clutter. According to the Euclidean mean, the authors combined an available prior covariance matrix with the persymmetric structure covariance estimator, symmetric structure covariance estimator, and Toeplitz structure covariance estimator, respectively, to derive three knowledge-aided structured covariance estimators. At the analysis stage, the authors assess the performance of the proposed estimators in estimation accuracy and detection probability. The analysis is conducted both on the simulated data and real sea clutter data collected by the IPIX radar sensor system. The results show that the knowledge-aided Toeplitz structure covariance estimator (KA-T) has the best performance both in estimation and detection, and the knowledge-aided persymmetric structure covariance estimator (KA-P) has similar performance with the knowledge-aided symmetric structure covariance estimator (KA-S). Moreover, compared with existing knowledge-aided estimator, the proposed estimators can obtain better performance when secondary data are insufficient.

Matrix versions of the Hellinger distance

On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$ where $\mathcal{A}(A,B)$ is the arithmetic mean and $\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When $\mathcal{G}(A,B)=A^{1/2}B^{1/2}$ this distance is $\|A^{1/2}-B^{1/2}\|_2,$ and when $\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$ it is the Bures-Wasserstein metric. We study two other cases: $\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ the Pusz-Woronowicz geometric mean, and $\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$ the log Euclidean mean. With these choices $d(A,B)$ is no longer a metric, but it turns out that $d^2(A,B)$ is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of $m$ positive definite matrices with respect to these distance measures.

Examination of Mean Stress Calculation Approaches in Rock Mechanics

The mean stress, as a fundamental statistical property of a group of stress data, is essential for stress variability characterisation. However, currently in rock mechanics, the mean stress is customarily and erroneously calculated by separately averaging the principal stress magnitudes and orientations. In order to draw the attention of our community to the appropriate approach for stress variability characterisation, here we compare the customary scalar/vector mean with that obtained by the mathematically rigorous tensorial approach—the Euclidean mean. Calculation of mean stress using both a small group of actual in situ stress measurement results and a large group of simulated stress data (obtained using the combined finite–discrete element method, FEMDEM) demonstrates that the two approaches yield different results. Further investigation of these results shows that the scalar/vector approach may yield non-unique and non-orthogonal mean principal stresses, and these may deviate significantly from those of the Euclidean mean. Our calculations and comparisons reveal that the scalar/vector approach is deficient because it processes the principal stress magnitudes and orientations separately as independent quantities and ignores the connection between them. Conversely, the tensorial approach appropriately averages the tensors that simultaneously carry not only the information of stress magnitudes and orientations, but also the inherent relations between them. Therefore, arbitrarily using scalar/vector mean stress of in situ stress measurements as input in further rock engineering analyses may yield significantly erroneous results. We advise that stress data should be statistically processed in a tensorial manner using tensors referred to a common Cartesian coordinate system.

Inequalities for the Wasserstein mean of positive definite matrices

We prove majorization inequalities for different means of positive definite matrices. These include the Cartan mean (the Karcher mean), the log Euclidean mean, the Wasserstein mean and the power mean.

Investigating the Relationship Between Far-Field Stress and Local Values of the Stress Tensor

In situ stress is an important parameter in rock mechanics, thus robust estimation of far–field stress to be used as boundary loadings for further rock engineering analysis based on the local in situ stress data seems indispensable. Here, as part of a preliminary investigation into this problem, we use the combined finite–discrete element method to examine how the mean of local stress tensors is related to the far–field stress. We have conducted a series of stress simulations on a model of a fractured rock mass subjected to various boundary loadings, and calculated the Euclidean mean of the stress data and compared them with the boundary loadings. The results shows that the Euclidean mean and boundary loadings are approximately equal, which gives us an indication that the Euclidean mean of the stress data can be a reasonable estimation of the far–field stress.

An iterative denoising system based on Wiener filtering with application to biomedical images

Biomedical image denoising systems are important for accurate clinical diagnosis. The purpose of this study is to present a simple and effective iterative multistep image denoising system based on Wiener filtering (WF) where the denoised image from one stage is the input to the next stage. The denoising process stops when a particular condition measured by image energy is adaptively achieved. The proposed iterative system is tested on real clinical images and performance is measured by the well-known peak-signal-to-noise-ratio (PSNR) statistic. Experimental results showed that the proposed iterative system outperforms conventional image denoising algorithms; including wavelet packet (WP), fourth order partial differential equation (FOPDE), nonlocal Euclidean means (NLEM), first order local statistics (FOLS), and single Wiener filter used as baseline model. The experimental results demonstrate that the proposed approach can remove noise automatically and effectively while edges and texture characteristics are preserved.

Cyclotomy and the heptadecagon

Cyclotomy is concerned with the division of a circle into a given number of equal segments, amounting to the construction of a regular polygon, say a q-gon, so that we need to deliver the angle α = 2π / q, or the length cos α. The construction is by Euclidean means, which make use of only ruler and compasses. Now, from given lengths, sums and differences of lengths are easy to obtain and, with the compasses, products and quotients of lengths can be obtained from similar triangles using parallel lines; indeed even the length can be obtained by applying the intersecting chord theorem to a circle with diameter 1 + a. However, there is not much else one can do with the compasses, so that the length cos α has to come from the real roots of a sequence of quadratic equations with ‘suitable’ coefficients — the meaning of being suitable will be made clear later.Gauss made the first significant contribution to the classical theory of cyclotomy in Article 365 of his famous Disquisitiones Arithmeticae [1] in 1801. He showed that the construction is possible if q = p is a Fermat prime, that is a prime of the form 22n + 1; see §7 for a necessary and sufficient condition for q. The only known Fermat primes are p = 3, 5, 17, 257, 65537; the cases p = 3, 5 and 17 correspond to the construction of the equilateral triangle, the regular pentagon, and the regular heptadecagon, the details for which Gauss gave.

Mean and dispersion of stress tensors using Euclidean and Riemannian approaches

Stress is central to many aspects of rock mechanics, and in the analysis of in situ stress measurement data the calculation of the mean value and an assessment of dispersion are important for statistical characterisation. Currently, stress magnitude and orientation are processed separately in such analyses. This effectively decomposes the second-order stress tensor into scalar (principal stress magnitudes) and vector (principal stress orientations) components, and calculation of mean and dispersion of stress data on the basis of these decomposed components, which violates the tensorial nature of stress, may either yield biased results or be difficult to conduct. Here, by introducing tensorial techniques, we examine two calculation approaches for the mean and dispersion for stress tensors – based on Euclidean and Riemannian geometries – and discuss their similarities, differences and potential applicability in engineering practice. We compare the two approaches using stress tensor superposition and interpolation, and the analysis of actual in situ stress data. The results indicate that Euclidean and Riemannian mean tensors are in general not equal, with the disparity increasing as stress tensor dispersion increases. Both Euclidean and Riemannian approaches are shown to be capable of characterising stress dispersion, although Euclidean dispersion is scale dependent and has units of stress whereas Riemannian dispersion is a scale independent unitless number. Finally, a paradox is revealed in that despite stress tensors being Riemannian entities, it is Euclidean mean stress that is the more meaningful for engineering applications.

Using moving North Pacific index to improve rainy season rainfall forecast over the Yangtze River basin by analog error correction

A new analog error correction (AEC) scheme based on the moving North Pacific index (MNPI) is designed in this study. This scheme shows obvious improvement in the prediction skill of the operational coupled general circulation model (CGCM) of the National Climate Center of China for the rainy season rainfall (RSR) anomaly pattern correlation coefficient (ACC) over the mid-to-lower reaches of the Yangtze River (MLRYR). A comparative analysis indicates that the effectiveness of the new scheme using the MNPI is better than the system error correction scheme using the North Pacific index (NPI). A Euclidean distanceweighted mean rather than a traditional arithmetic mean, is applied to the integration of the analog year’s prediction error fields. By using the MNPI AEC scheme, independent sample hindcasts of RSR during the period 2003–2009 are then evaluated. The results show that the new scheme exhibited a higher forecast skill during 2003–2009, with an average ACC of 0.47; while the ACC for the NPI case was only 0.19. Furthermore, the forecast skill of the RSR over the MLRYR is examined. In the MNPI case, empirical orthogonal function (EOF) was used in the degree compression of the prediction error fields from the CGCM, whereas the AEC scheme was applied only to its first several EOF components for which the accumulative explained variance accounted for 80% of the total variance. This further improved the ACC of the independent sample hindcasts to 0.55 during the 7-yr period.

On the measure of the cut locus of a Fréchet mean

One of the fundamental differences between the Central Limit Theorem for empirical Frechet means obtained in [Kendall and Le, ‘Limit theorems for empirical Frechet means of independent and non‐identically distributed manifold‐valued random variables’, Braz. J. Probab. Stat. 25 (2011) 323–352] and that for empirical Euclidean means lies on the assumption that the probability measure of the cut locus of the true Frechet mean is zero. In [Hotz and Huckemann, ‘Intrinsic means on the circle: uniqueness, locus and asymptotics’, Preprint, 2011, arXiv:1108.2141v1], the authors show that, in the case of a circle, this assumption holds automatically. This paper shows that this holds for any complete and connected Riemannian manifold, assuming that there are at least two minimal geodesics between the Frechet mean and any point in its cut locus and that, in fact, it can also be generalized to local minima of the