Tensor Analysis Research Articles

Galvanic distortion decomposition of magnetotelluric impedance tensors in 1-D electrical anisotropic media

SUMMARY The electromagnetic (EM) local distortion of the transfer function due to shallow small-scale inhomogeneities hinders the accurate interpretation of magnetotelluric (MT) data. Under the assumption that regional subsurface structures are electrically isotropic, decomposition techniques of the MT impedance tensor that focus on the galvanic distortion of the electric field have been well developed. In this paper, we present a decomposition method of MT impedance tensors over a regional 1-D conductivity anisotropic Earth, in which the galvanic distortion of both the electric and the associated magnetic fields are taken into account. An eigenparameter analysis is introduced to evaluate the intrinsic indeterminacy of magnetic distortion parameters. Regional anisotropic responses and EM distortion parameters are resolved by using a modified BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton algorithm combined with phase tensor analysis and the trust region method. In the presence of near-surface anisotropic inhomogeneities, synthetic 2-D data show an accentuated magnetic galvanic effect in the response tensors especially in the diagonal components probably due to the infinite extension in the strike direction. However, the magnetic galvanic distortion is not significant in synthetic 3-D models. The decomposition scheme is also applied to the analysis of the BC87 data set collected in southeastern British Columbia to investigate both the electric and magnetic field galvanic distortion.

Impact of the Offshore Seismograph Network and 3‐D Seismic Velocity Structure Model on Centroid Moment Tensor Analysis for Offshore Earthquakes: Application to the Japan Trench Subduction Zone

Impact of the Offshore Seismograph Network and 3‐D Seismic Velocity Structure Model on Centroid Moment Tensor Analysis for Offshore Earthquakes: Application to the Japan Trench Subduction Zone

Cryo-X-Ray Phase Contrast Imaging Enables Combined 3D Structural Quantification and Nucleic Acid Analysis of Myocardial Biopsies.

Snap-frozen biopsies serve as a valuable clinical resource of archival material for disease research, as they enable a comprehensive array of downstream analyses to be performed, including extraction and sequencing of nucleic acids. Obtaining three-dimensional (3D) structural information before multi-omics is more challenging but can potentially allow for better characterization of tissues and targeting of clinically relevant cells. Conventional histological techniques are limited in this regard due to their destructive nature and the reconstruction artifacts produced by sectioning, dehydration, and chemical processing. These limitations are particularly notable in soft tissues such as the heart. In this study, the feasibility of using synchrotron-based cryo-X-ray phase contrast imaging (cryo-X-PCI) of snap-frozen myocardial biopsies is assessed and 3D structure tensor analysis of aggregated myocytes, followed by nucleic acid (DNA and RNA) extraction and analysis. It is shown that optimal sample preparation is the key driver for successful structural and nucleic acid preservation which is unaffected by the process of cryo-X-PCI. It is proposed that cryo-X-PCI has clinical value for 3D tissue analysis of cardiac and potentially non-cardiac soft tissue biopsies before nucleic acid investigation.

Landau vs Einstein: mathematics represents the universe

The modern theory of gravity is the Einstein space theory, which is described by mathematical methods of tensor analysis of Riemannian geometry. Einstein built on this basis one of the most successful and amazing theories. Complexity theories often lead to the extraction of foundations and, as is customary in theory, to obtaining meaningless results. Now the entire intellectual power of numerous researchers is directed at alternative theories, often based on erroneous ideas about Einstein's theory. An example of this is the catastrophic error in the definition of the coordinate transformation in the top manual on theoretical physics by L. D. Landau and E. M. Lifshitz.

Long ranged stress correlations in the hard sphere liquid.

The smooth emergence of shear elasticity is a hallmark of the liquid to glass transition. In a liquid, viscous stresses arise from local structural rearrangements. In the solid, Eshelby has shown that stresses around an inclusion decay as a power law r-D, where D is the dimension of the system. We study glass-forming hard sphere fluids by simulation and observe the emergence of the unscreened power-law Eshelby pattern in the stress correlations of the isotropic liquid state. By a detailed tensorial analysis, we show that the fluctuating force field, viz., the divergence of the stress field, relaxes to zero with time in all states, while the shear stress correlations develop spatial power-law structures inside regions that grow with longitudinal and transverse sound propagation. We observe the predicted exponents r-D and r-D-2. In Brownian systems, shear stresses relax diffusively within these regions, with the diffusion coefficient determined by the shear modulus and the friction coefficient.

Audio completion method based on tensor analysis

Audio completion method based on tensor analysis

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Sedimentation dynamics of triply twisted Möbius bands: Geometry versus topology

We explore the sedimentation dynamics of triply twisted Möbius bands, each characterized by threefold rotational symmetry but distinguished by its construction and intrinsic geometrical properties. Three types of bands are considered: one with vanishing Gaussian curvature, constructed by isometrically deforming flat rectangular strips through bending without stretching; and two with negative Gaussian curvature, one being constructed by isometrically deforming helicoids. Experiment on these three types of bands, with a focus on varying aspect ratios, reveals a singular phenomenon: while the spin directions of bands not derived from helicoids spin in directions consistent with their inherent chirality, bands derived from helicoids exhibit an aspect-ratio-dependent spin, pointing to the existence of a critical aspect ratio at which geometric factors dominate over chiral influences. Supported by numerical simulations and a detailed analysis of the resistance tensors, we propose the unique response of bands derived from helicoids originates from a complex interaction among geometry, topology, and hydrodynamics. Two explanations are offered for the chiral transition observed in bands derived from helicoid. First, this transition may parallel the dynamics of superhelices, for which competing chiralities influence rotational behavior. Second, the unique geometrical properties of bands derived from helicoids, coupled with deviations between the rotation axis and the local symmetry axis, may underlie the observed aspect-ratio-dependent chiral transition. Our study underscores the significant role of geometrical and topological nuances in determining the behavior of chiral objects suspended in fluids. In addition to offering transformative potential across diverse fields, it promises advancements in mixing, separation processes, and innovative passive swimmers. Published by the American Physical Society 2024

Micron-resolution fiber mapping in histology independent of sample preparation.

Detailed knowledge of the brain's nerve fiber network is crucial for understanding its function in health and disease. However, mapping fibers with high resolution remains prohibitive in most histological sections because state-of-the-art techniques are incompatible with their preparation. Here, we present a micron-resolution light-scattering-based technique that reveals intricate fiber networks independent of sample preparation for extended fields of view. We uncover fiber structures in both label-free and stained, paraffin-embedded and deparaffinized, newly-prepared and archived, animal and human brain tissues - including whole-brain sections from the BigBrain atlas. We identify altered microstructures in demyelination and hippocampal neurodegeneration, and show key advantages over diffusion magnetic resonance imaging, polarization microscopy, and structure tensor analysis. We also reveal structures in non-brain tissues - including muscle, bone, and blood vessels. Our cost-effective, versatile technique enables studies of intricate fiber networks in any type of histological tissue section, offering a new dimension to neuroscientific and biomedical research.

The effects of thickness, polarization, and strain on vibrational modes of 2D Fe3GeTe2

In this study, we investigated the effects of thickness, light polarization, and strain on the Raman spectra of two-dimensional (2D) Fe3GeTe2 (FGT) crystals synthesized via chemical vapor transport. The crystals are thoroughly characterized using a combination of microscopic, diffraction, and spectroscopic techniques. Particularly, a systematic angle-resolved polarized Raman spectroscopy study reveals a clear polarization dependence of the Raman intensity in both parallel and crossed polarization directions, with the Ag1 mode completely disappearing in the crossed polarization direction. The angle-dependent intensity of both the Ag1 and E2g2 modes in parallel polarization and the intensity of the E2g2mode in the crossed polarization remain constant at all angles. These findings align with predictions from Raman tensor analysis, providing compelling evidence for the unambiguous assignment of the Ag1 and E2g2 modes to specific peaks observed in the Raman spectrum of FGT, resolving existing confusion in the literature regarding their assignment. Furthermore, we examined the effect of strain on the Raman spectrum of 2D FGT in-situ using a bending device. Our study, conducted on a monolayer to few-layer 2D FGT deposited onto polyethylene terephthalate and subjected to outward (inward), i.e., tensile (compressive) bending, demonstrates appreciable downshifting (upshifting) of the Raman peak position of both Ag1 and E2g2 modes. These findings are particularly significant given that strain engineering represents an effective approach to modulate the magnetic properties of FGT and other 2D magnetic materials.

Electrostatic Surface Potentials and Chalcogen-Bonding Motifs of Substituted 2,1,3-Benzoselenadiazoles Probed via 77Se Solid-State NMR Spectroscopy.

Chalcogen bonds (ChB) are moderately strong, directional, and specific non-covalent interactions that have garnered substantial interest over the last decades. Specifically, the presence of two σ-holes offers great potential for crystal engineering, catalysis, biochemistry, and molecular sensing. However, ChB applications are currently hampered by a lack of methods to characterize and control chalcogen bonds. Here, we report on the influence of various substituents (halogens, cyano, and methyl groups) on the observed self-complementary ChB networks of 2,1,3-benzoselenadiazoles. From molecular electrostatic potential calculations, we show that the electrostatic surface potentials (ESP) of the σ-holes on selenium are largely influenced by the electron-withdrawing character of these substituents. Structural analyses via X-ray diffraction reveal a variety of ChB geometries and binding modes that are rationalized via the computed ESP maps, although the structure of 5,6-dimethyl-2,1,3-benzoselenadiazole also demonstrates the influence of steric interactions. 77Se solid-state magic-angle spinning NMR spectroscopy, in particular the analysis of the selenium chemical shift tensors, is found to be an effective probe able to characterize both structural and electrostatic features of these self-complementary ChB systems. We find a positive correlation between the value of the ESP maxima at the σ-holes and the experimentally measured 77Se isotropic chemical shift, while the skew of the chemical shift tensor is established as a metric which is reflective of the ChB binding motif.

Robust Corrupted Data Recovery and Clustering via Generalized Transformed Tensor Low-Rank Representation.

Tensor analysis has received widespread attention in high-dimensional data learning. Unfortunately, the tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. How to recover the characteristics of the corrupted tensor data and make it compatible with the downstream clustering task remains a challenging problem. In this article, we study a generalized transformed tensor low-rank representation (TTLRR) model for simultaneously recovering and clustering the corrupted tensor data. The core idea is to find the latent low-rank tensor structure from the corrupted measurements using the transformed tensor singular value decomposition (SVD). Theoretically, we prove that TTLRR can recover the clean tensor data with a high probability guarantee under mild conditions. Furthermore, by using the transform adaptively learning from the data itself, the proposed TTLRR model can approximately represent and exploit the intrinsic subspace and seek out the cluster structure of the tensor data precisely. An effective algorithm is designed to solve the proposed model under the alternating direction method of multipliers (ADMMs) algorithm framework. The effectiveness and superiority of the proposed method against the compared methods are showcased over different tasks, including video/face data recovery and face/object/scene data clustering.

Integral theorems for the gradient of a vector field, with a fluid dynamical application

The familiar divergence and Kelvin–Stokes theorem are generalized by a tensor-valued identity that relates the volume integral of the gradient of a vector field to the integral over the bounding surface of the tensor product of the vector field with the exterior normal. The importance of this long-established yet relatively little-known result is discussed. In flat two-dimensional space, it reduces to a relationship between an integral over an area and that over its bounding curve, combining the two-dimensional divergence and Kelvin–Stokes theorems together with two related theorems involving the strain, as is shown through a decomposition using a suitable tensor basis. A fluid dynamical application to oceanic observations along the trajectory of a moving platform is given. The potential extension of the generalized two-dimensional identity to curved surfaces is considered and is shown not to hold. Finally, the paper includes a substantial background section on tensor analysis, and presents results in both symbolic notation and index notation in order to emphasize the correspondence between these two notational systems.

Dimensionality and Geoelectric Strike Analysis of 2D Magnetotelluric Modeling in Geothermal Area “XYZ”

Abstract Magnetotelluric (MT) method is crucial in geothermal exploration due to its capability to delineate geothermal prospect area by mapping subsurface resistivity distribution. However, in 2D Magnetotelluric modelling, galvanic distortion can distort the regional MT responses, leading to interpretation errors. This study employs phase tensor analysis to eliminate galvanic distortion. Results reveal the dimensionality of each station and three variations of the tipper strike, often referred to as the geoelectric strike. The tipper strike of each station showed varying values, while the tipper strike of all stations indicated N35°E. The tipper strike results required validation against additional geological information, such as faults, fractures, and changes in lithology, significantly influence the conductivity structure of subsurface, which indicated a geological strike of approximately N30°E. This study compares the MT data without rotation, rotation to tipper strike of each station, and tipper strike of all stations to assess effectiveness in eliminating galvanic distortion and improving data quality. The resulting 2D Magnetotelluric cross-section rotated to the tipper strike of all station, exhibits the lowest RMS Error of 2% and roughness of 654.6 among other variations, indicating improve quality. This interpretation identifies a geothermal system in the study area, closely aligning with actual subsurface geological conditions.

Strain rate effects on fragment morphology of ceramic alumina: A synchrotron-based study

Dynamic (600–1000 s−1) and quasi-static (0.001–0.01 s−1) compression experiments are conducted on a high-purity alumina ceramic using a split Hopkinson pressure bar and a material test system, respectively. The postmortem fragments of ceramic samples at different strain rates are then characterized via synchrotron micro computed tomography (CT). The three-dimensional (3D) morphologies of fragments are quantified using the gyration tensor analysis after proper segmentation of CT images. The mean fragment size decreases in a power-law form while the mean shape indices (sphericity, elongation index and flatness index) increase in a logarithmic-linear form, with increasing strain rates. In addition, the fragment size and shape distributions are all found to follow the Weibull probability distribution. To reveal the underlying mechanisms of such strain rate effects, high-speed optical and X-ray imaging are implemented to capture the fracture process of ceramic samples under quasi-static and dynamic compression. Primary and secondary wing cracks (PWCs/SWCs) control the fracture and fragmentation of the ceramic under both loading conditions. Compared to quasi-static loading, a considerably larger number of SWCs are produced under dynamic loading, and pronounced branching and bridging occur among the PWCs, which prevent the PWCs from coalescing into axial splitting cracks. Consequently, crack networks composed of high-density wing cracks break the sample into finer and more isotropic fragments, consistent with CT characterizations. Scanning electron microscopy is utilized to analyze the micro damage modes of ceramic samples. Transgranular fracture dominates the grain-scale damage and contributes to the higher dynamic fracture resistance of the alumina under dynamic loading, as a result of more homogeneous nucleation and growth of micro cracks.

Advanced Hyperspectral Image Analysis: Superpixelwise Multiscale Adaptive T-HOSVD for 3D Feature Extraction.

Hyperspectral images (HSIs) possess an inherent three-order structure, prompting increased interest in extracting 3D features. Tensor analysis and low-rank representations, notably truncated higher-order SVD (T-HOSVD), have gained prominence for this purpose. However, determining the optimal order and addressing sensitivity to changes in data distribution remain challenging. To tackle these issues, this paper introduces an unsupervised Superpixelwise Multiscale Adaptive T-HOSVD (SmaT-HOSVD) method. Leveraging superpixel segmentation, the algorithm identifies homogeneous regions, facilitating the extraction of local features to enhance spatial contextual information within the image. Subsequently, T-HOSVD is adaptively applied to the obtained superpixel blocks for feature extraction and fusion across different scales. SmaT-HOSVD harnesses superpixel blocks and low-rank representations to extract 3D features, effectively capturing both spectral and spatial information of HSIs. By integrating optimal-rank estimation and multiscale fusion strategies, it acquires more comprehensive low-rank information and mitigates sensitivity to data variations. Notably, when trained on subsets comprising 2%, 1%, and 1% of the Indian Pines, University of Pavia, and Salinas datasets, respectively, SmaT-HOSVD achieves impressive overall accuracies of 93.31%, 97.21%, and 99.25%, while maintaining excellent efficiency. Future research will explore SmaT-HOSVD's applicability in deep-sea HSI classification and pursue additional avenues for advancing the field.

Investigation of macro-micro mechanical behaviors and failure mechanisms in cemented tailings backfill with varying proportions of fine

This study investigates the impact of tailings characteristics, particularly fine tailings, on the efficiency and effectiveness of mining engineering backfill operations. Contrary to the common underestimation, fine tailings significantly affect the mechanical properties and failure mechanisms of Cemented Tailings Backfill (CTB). Through a series of experiments examining various particle size distributions and employing advanced Scanning Electron Microscopy (SEM) and Mercury Intrusion Porosimetry (MIP) techniques, this research clarifies the role of fine tailings in modifying the mechanical strength and failure patterns of CTB. Adding 20 % fine tailings optimally enhances both the early and long-term strength of backfill materials. With aging, adding 30 % fine tailings shows a rapid strength increase from 7d to 28d, with the 28d strength slightly lower than that of specimens with 20 % addition, but the specimens demonstrate better ductility after reaching peak strength. As the content of fine tailings increases, crack propagation in the specimens shifts from radial to axial, simplifying the fracture mode from complex to singular. Consequently, the fracture mechanism changes from ductile to brittle, and then reverts to ductile. Moreover, using particle flow simulation and moment tensor analysis to study the failure processes indicates that while increased fine particle content boosts the material's initial strength, it ultimately leads to a simpler crack network and volumetric expansion as the primary failure mode. In terms of energy conversion mechanisms, as the content of fine particles increases, the energy first increases and then decreases. During the elastic stage and the early stage of the plastic stage, the energy is mainly converted into strain energy and PB bonding energy, which affects the elasticity of the sample. In the later stage of the plastic phase, the energy is mainly converted into dissipated energy, leading to the failure of the sample.

Matrix methods and teaching the fundamentals of fluid and gas mechanics in higher education institutions

Problem. The instruction of physical and mathematical disciplines in higher education institutions employing cutting-edge methods and universal approaches, which seamlessly integrate scientific and engineering activities of future specialists, remains one of the pertinent and priority tasks. The utilization of multidimensional or even infinite-dimensional spaces has become an effective and nearly indispensable tool in mathematical modeling of physical phenomena. This circumstance is directly linked to the increasing level of abstraction and the refinement of mathematical methods. The use of geometric methods inherent in vector algebra and vector analysis as the foundation for studying mechanics, despite their illustrative nature, is losing its relevance. Methods built upon matrix formalism are evolving as substitutes. The matrix framework enables the exploitation of phase space advantages to derive canonical equations of motion for continuous media and electromagnetic field equations in covariant forms. The electromagnetic potential acquires mechanical significance, allowing the utilization of electromechanical analogies at a fundamental level rather than on a merely formal basis, as done in classical electrodynamics. Goal. The aim of this research is to study and justify the advantages of matrix methods for deriving equations of motion for fluids and gases from the canonical equations of mechanics by utilizing a continuum model in phase and physical space. The research is directed towards demonstrating the connection between the internal microscopic and macroscopic motion of fluids and gases, as well as highlighting the potential of matrix formalism both in terms of modeling and the utilization of computational tools. Methodology. The methodological basis for selecting matrix methods lies in the application of tensor analysis and its generalizations for modeling the motion of fluids and gases in physical and phase space. Results. It has been demonstrated that the matrix method, previously applied to classical and relativistic mechanics, allows for the consideration of the molecular structure of the environment by folding the multidimensional phase space of the mechanical system, represented by a particle of the medium, and thereby deriving equations of motion for fluids and gases from the canonical equations of classical mechanics and the equations of momentum balance of the medium. Originality. The combination of canonical equations as a result of applying the conservation law of matter in the form of balance equations in phase space and momentum balance equations in physical space allows for elucidating the relationship between macroscopic and microscopic motions of the medium, as well as the mathematical structure of the equations of motion for fluids and gases. Practical value. The proposed method allows, while remaining within the standard mathematical training offered by technical educational institutions, to effectively formalize the equations of motion for fluids and gases and provide them with an invariant form. By guiding oneself through invariance, both fundamental laws (such as the law of universal gravitation) and partial laws (the generalized Newton's law) can be obtained.

Identification of cancer risk groups through multi-omics integration using autoencoder and tensor analysis.

Identifying cancer risk groups by multi-omics has attracted researchers in their quest to find biomarkers from diverse risk-related omics. Stratifying the patients into cancer risk groups using genomics is essential for clinicians for pre-prevention treatment to improve the survival time for patients and identify the appropriate therapy strategies. This study proposes a multi-omics framework that can extract the features from various omics simultaneously. The framework employs autoencoders to learn the non-linear representation of the data and applies tensor analysis for feature learning. Further, the clustering method is used to stratify the patients into multiple cancer risk groups. Several omics were included in the experiments, namely methylation, somatic copy-number variation (SCNV), micro RNA (miRNA) and RNA sequencing (RNAseq) from two cancer types, including Glioma and Breast Invasive Carcinoma from the TCGA dataset. The results of this study are promising, as evidenced by the survival analysis and classification models, which outperformed the state-of-the-art. The patients can be significantly (p-value<0.05) divided into risk groups using extracted latent variables from the fused multi-omics data. The pipeline is open source to help researchers and clinicians identify the patients' risk groups using genomics.

Study on structural, morphological, elastic and electrical properties of ZnO nanoparticles for electronic device applications

The present study investigates the synthesis and analysis of zinc oxide nanoparticles (ZnO NPs) intended for use in electronic devices. ZnO NPs with an average crystallite size of 33 nm and a hexagonal wurtzite structure were produced by the combustion process. Scanning Electron Microscopy (SEM) revealed a densely packed, spherical morphology. Elastic properties, studied through the General Utility Lattice Program (GULP) and Elastic Tensor Analysis (ELATE), reveal the ductile nature of ZnO NPs. Optical studies exhibited a characteristic absorption peak at 366 nm, with a calculated optical band gap of 2.36 eV. Additionally, efforts are being made to determine refractive index (n), electronic polarizability (α) and optical susceptibility (χ). Electrical properties, including dielectric behaviour, AC conductivity, and I–V characteristics, demonstrated the influence of relaxation phenomena and revealed non-ohmic conduction. The dielectric studies show that dielectric constant, dielectric loss, and impedance are frequency and temperature dependent. Richardson-Schottky emission was identified as the predominant conduction mechanism, highlighting the potential of ZnO NPs in electronic devices.