Spatial Tensor Research Articles

The use of principal spatial tensor to predict anisotropic fiber orientation in concentrated fiber suspensions

The Phelps-Tucker anisotropic rotary diffusion (ARD) equation constitutes an important development in relation to concentrated fiber suspension rheology and is significant in its ability to predict transient, flow-induced fiber orientation, especially for long fiber composites in the industry. Within this model, a critical spatial tensor was assumed to be a polynomial tensor-valued function depending on both the orientation tensor and the rate-of-strain tensor. However, it can be difficult to derive a large number of fitting ARD parameters. For simplification, we newly defined the spatial tensor as coaxial with the orientation tensor's principal directions, while the spatial tensor's principal components are to control anisotropic changes in the orientation tensor. Such a principal spatial tensor used in the Phelps-Tucker ARD model is demonstrated in accurately predicting the shell-core structure of fiber orientation distributions for injection molding simulation of long fiber composites, supported with e...

Covariant Hamiltonian tetrad approach to numerical relativity

A Hamiltonian approach to the equations of general relativity is proposed using the powerful mathematical language of multivector-valued differential forms. In the approach, the gravitational coordinates are the 12 spatial components of the line interval (the vierbein) including their antisymmetric parts, and their 12 conjugate momenta. A feature of the proposed formalism is that it allows Lorentz gauge freedoms to be imposed on the Lorentz connections rather than on the vierbein, which may facilitate numerical integration in some challenging problems. The 40 Hamilton's equations comprise 12 + 12 = 24 equations of motion, 10 constraint equations (first class constraints, which must be arranged on the initial hypersurface of constant time, but which are guaranteed thereafter by conservation laws), and 6 identities (second class constraints). The 6 identities define a trace-free spatial tensor that is the gravitational analog of the magnetic field of electromagnetism. If the gravitational magnetic field is promoted to an independent field satisfying its own equation of motion, then the system becomes the WEBB system, which is known to be strongly hyperbolic. Some other approaches, including ADM, BSSN, WEBB, and Loop Quantum Gravity, are translated into the language of multivector-valued forms, bringing out their underlying mathematical structure.

A geometric rationale for invariance, covariance and constitutive relations

There are, in each branch of science, statements which, expressed in ambiguous or even incorrect but seemingly friendly manner, were repeated for a long time and eventually became diffusely accepted. Objectivity of physical fields and of their time rates and frame indifference of constitutive relations are among such notions. A geometric reflection on the description of frame changes as spacetime automorphisms, on induced push–pull transformations and on proper physico–mathematical definitions of material, spatial and spacetime tensor fields and of their time-derivatives along the motion, is here carried out with the aim of pointing out essential notions and of unveiling false claims. Theoretical and computational aspects of nonlinear continuum mechanics, and especially those pertaining to constitutive relations, involving material fields and their time rates, gain decisive conceptual and operative improvement from a proper geometric treatment. Outcomes of the geometric analysis are frame covariance of spacetime velocity, material stretching and material spin. A univocal and frame-covariant tool for evaluation of time rates of material fields is provided by the Lie derivative along the motion. The postulate of frame covariance of material fields is assessed to be a natural physical requirement which cannot interfere with the formulation of constitutive laws, with claims of the contrary stemming from an improper imposition of equality in place of equivalence.

English

This paper determines contaminant dispersivity in the coastal sand of River Mersey Outer Estuary using results of field measurement from dye solute injection. Dispersivity is an alternative parameter for describing the characteristic behaviour of porous media. In this case, 3-D plumes in 2-D Photo-images were taken from multiple injection zones across the foreshore. The colour images were calibrated, standardized and the optical intensity converted into concentration maps. Apparent dispersivity of the subsurface foreshore was graphically determined from spatial covariance tensors. The variances were obtained by subtracting the products of the zeroth moments and square of the mean location of mass centers from the second order moments, and results were divided by the zeroth moment. The variance tensors (xʹxʹ2, and yʹyʹ2) were plotted against their respective travel distance intervals. The result predicts linear increase of the variance of spread distance, enhancing estimate of dispersivity using the best fit relations   and  for the x-x and y-y planes. The dispersivity coefficient varied with increase in travel distance. It quantifies the rate of spreading of the mean concentration of the tracer front which shows dependence on the distance traveled by the front. The effect on the mean concentration at the injection point is therefore described by the area of the effective dispersivity coefficient controls. Key words: Sand aquifers, porous media, injectate, vadose zone, ground water, transport of pollutant, contaminant transport.

Monitoring In-Vivo the Mammary Gland Microstructure during Morphogenesis from Lactation to Post-Weaning Using Diffusion Tensor MRI.

Lactation and the return to the pre-conception state during post-weaning are regulated by hormonal induced processes that modify the microstructure of the mammary gland, leading to changes in the features of the ductal / glandular tissue, the stroma and the fat tissue. These changes create a challenge in the radiological workup of breast disorder during lactation and early post-weaning. Here we present non-invasive MRI protocols designed to record in vivo high spatial resolution, T2-weighted images and diffusion tensor images of the entire mammary gland. Advanced imaging processing tools enabled tracking the changes in the anatomical and microstructural features of the mammary gland from the time of lactation to post-weaning. Specifically, by using diffusion tensor imaging (DTI) it was possible to quantitatively distinguish between the ductal / glandular tissue distention during lactation and the post-weaning involution. The application of the T2-weighted imaging and DTI is completely safe, non-invasive and uses intrinsic contrast based on differences in transverse relaxation rates and water diffusion rates in various directions, respectively. This study provides a basis for further in-vivo monitoring of changes during the mammary developmental stages, as well as identifying changes due to malignant transformation in patients with pregnancy associated breast cancer (PABC).

Heliosheath Processes and the Structure of the Heliopause: Modeling Energetic Particles, Cosmic Rays, and Magnetic Fields

This paper summarizes the results obtained by the team "Heliosheath Processes and the Structure of the Heliopause: Modeling Energetic Particles, Cosmic Rays, and Magnetic Fields" supported by the International Space Science Institute in Bern, Switzerland. We focus on the physical processes occurring in the outer heliosphere, especially at its boundary called the heliopause (HP), and in the LISM. The importance of magnetic field, charge exchange between atoms and ions, and solar cycle on the heliopause topology and observed heliocentric distances to different heliospheric discontinuities are discussed. It is shown that time-dependent boundary conditions are necessary to describe the heliospheric asymmetries detected by the Voyager spacecraft. We also discuss the structure of the HP, especially due to its instability and magnetic reconnection. It is demonstrated that the Rayleigh-Taylor instability of the nose of the HP creates consecutive layers of the interstellar and heliospheric plasma which are magnetically connected to different sources. This may be a possible explanation of abrupt changes in the galactic and anomalous cosmic ray fluxes observed by Voyager 1 when it was crossing the HP structure for a period of about one month in the summer of 2012. This paper also discusses the plausibility of fitting simulation results to a number of observational data sets obtained by in situ and remote measurements. The distribution of magnetic field in the vicinity of the HP is discussed in the context of Voyager measurements. We discuss the transport of energetic particles in the inner and outer heliosheath, concentrating on the anisotropic spatial diffusion diffusion tensor and the pitch-angle dependence of perpendicular diffusion and demonstrate that the latter can explain the observed pitch-angle anisotropies of both the anomalous and galactic cosmic rays in the outer heliosheath.

High-resolution computed microtomography for the characterization of a diffusion tensor imaging phantom

This paper addresses the issue of the quantitative characterization of the structure of the calibration model (phantom) for b-matrix spatial distribution diffusion tensor imaging (BSD-DTI) scanners. The aim of this study was to verify manufacturing assumptions of the structure of materials, since phantoms are used for BSD-DTI calibration directly after manufacturing. Visualization of the phantoms’ structure was achieved through optical microscopy and high-resolution computed microtomography (µCT). Using µCT images, a numerical model of the materials structure was developed for further quantitative analysis. 3D image characterization was performed to determine crucial structural parameters of the phantom: porosity, uniformity and distribution of equivalent diameter of capillary bundles. Additionally calculations of hypothetical flow streamlines were also performed based on the numerical model that was developed. The results obtained in this study can be used in the calibration of DTI-BST measurements. However, it was found that the structure of the phantom exhibits flaws and discrepancies from the assumed geometry which might affect BSD-DTI calibration.

Is Spacetime as Physical as Is Space?

Two questions are investigated by looking successively at classical mechanics, special relativity, and relativistic gravity: first, how is space related with spacetime? The proposed answer is that each given reference fluid, that is a congruence of reference trajectories, defines a physical space. The points of that space are formally defined to be the world lines of the congruence. That space can be endowed with a natural structure of 3-D differentiable manifold, thus giving rise to a simple notion of spatial tensor --- namely, a tensor on the space manifold. The second question is: does the geometric structure of the spacetime determine the physics, in particular, does it determine its relativistic or preferred-frame character? We find that it does not, for different physics (either relativistic or not) may be defined on the same spacetime structure --- and also, the same physics can be implemented on different spacetime structures. Keywords: Affine space; classical mechanics; special relativity; relativistic gravity; reference fluid.

Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain [Formula: see text] is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length [Formula: see text], we prove that the solution of any member of our family of effective equations is [Formula: see text]-close to the true oscillatory wave over a time interval of length [Formula: see text] in a norm equivalent to the [Formula: see text] norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

Weak* tensor products for von Neumann algebras

The category of $C^*$-algebras is blessed with many different tensor products. In contrast, virtually the only tensor product ever used in the category of von Neumann algebras is the normal spatial tensor product. We propose a definition of what a generic tensor product in this category should be. We call these weak* tensor products. For von Neumann algebras $M$ and $N$, there are, in general, many choices of weak* tensor completions of the algebraic tensor product $M\odot N$. In fact, we demonstrate that $M$ has the property that $M\odot N$ has a unique weak* tensor product completion for every von Neumann algebra $N$ if and only if $M$ is completely atomic, i.e., is a direct product of type I factors. This in particular implies that even abelian von Neumann algebras need not have this property. As an application of the theory developed throughout the paper, we construct $2^{\mathfrak c}$ nonequivalent weak* tensor product completions of $L^\infty(\mathbb R)\odot L^\infty(\mathbb R)$.

Diffusion tensor imaging in children with tuberous sclerosis complex: tract-based spatial statistics assessment of brain microstructural changes.

There is evidence of microstructural changes in normal-appearing white matter of patients with tuberous sclerosis complex. To evaluate major white matter tracts in children with tuberous sclerosis complex using tract-based spatial statistics diffusion tensor imaging (DTI) analysis. Eight children (mean age ± standard deviation: 8.5 ± 5.5 years) with an established diagnosis of tuberous sclerosis complex and 8 age-matched controls were studied. The imaging protocol consisted of T1-weighted high-resolution 3-D spoiled gradient-echo sequence and a spin-echo, echo-planar diffusion-weighted sequence. Differences in the diffusion indices were evaluated using tract-based spatial statistics. Tract-based spatial statistics showed increased axial diffusivity in the children with tuberous sclerosis complex in the superior and anterior corona radiata, the superior longitudinal fascicle, the inferior fronto-occipital fascicle, the uncinate fascicle and the anterior thalamic radiation. No significant differences were observed in fractional anisotropy, mean diffusivity and radial diffusivity between patients and control subjects. No difference was found in the diffusion indices between the baseline and follow-up examination in the patient group. Patients with tuberous sclerosis complex have increased axial diffusivity in major white matter tracts, probably related to reduced axonal integrity.

Taub-NUT as Bertrand Spacetime with Magnetic Fields

Based on symmetries Taub-NUT shares with Bertrand spacetime, we cast it as the latter with magnetic fields. Its nature as a Bianchi-IX gravitational instanton and other related geometrical properties are reviewed. We provide an easy derivation and comparison between the spatial Killing-Yano tensors deduced from first-integrals and the corresponding hyperk\"ahler structures and finally verify the existence of a graded Lie-algebra structure via Schouten-Nijenhuis brackets.

Spatial and material stress tensors in continuum mechanics of growing solid bodies

Spatial and material stress tensors in continuum mechanics of growing solid bodies

Innovative anisotropic phantoms for calibration of diffusion tensor imaging sequences

The paper describes a novel type of anisotropic phantoms designed for b-matrix spatial distribution diffusion tensor imaging (BSD-DTI). Cubic plate anisotropic phantom, cylinder capillary phantom and water reference phantom are described as a complete set necessary for calibration, validation and normalization of BSD-DTI.An innovative design of the phantoms basing on enclosing the anisotropic cores in glass balls filled with liquid made for the first time possible BSD calibration with usage of echo planar imaging (EPI) sequence. Susceptibility artifacts prone to occur in EPI sequences were visibly reduced in the central region of the phantoms.The phantoms were designed for usage in a clinical scanner's head coil, but can be scaled for other coil or scanner types. The phantoms can be also used for a pre-calibration of imaging of other types of phantoms having more specific applications.

The relation of spatial and tensor product of Arveson systems — the random set point of view

We characterise the embedding of the spatial product of two Arveson systems into their tensor product using the random set technique. An important implication is that the spatial tensor product does not depend on the choice of the reference units, i.e. it is an intrinsic construction. There is a continuous range of examples coming from the zero sets of Bessel processes where the two products do not coincide. The lattice of all subsystems of the tensor product is analysed in different cases. As a by-product, the Arveson systems coming from Bessel zeros prove to be primitive in the sense of Ref. 12.

Comparison theorems for causal diamonds

We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the centre. As a byproduct of our discussion we re-derive Bishop's inequality without assuming the positivity of the spatial Ricci tensor. We then generalize our considerations to arbitrary, static and not necessarily spherically symmetric, asymptotically flat spacetimes. In the case of spacetimes with a horizon our generalization involves the so-called {\it domain of dependence}. The respective volume, expressed in terms of the duration measured by a distant observer compared with the volume of the domain in Minkowski spacetime, exhibits behaviours which differ if $d=4$ or $d>4$. This peculiarity of four dimensions is due to the logarithmic subleading term in the asymptotic expansion of the metric near infinity. In terms of the invariant duration measured by a co-moving observer associated with the diamond we establish an inequality which is universal for all $d$. We suggest some possible applications of our results including the comparison theorems for entanglement entropy, causal set theory and fundamental limits on computation.

The b matrix calculation using the anisotropic phantoms for DWI and DTI experiments.

B-matrix Spatial Distribution Diffusion Tensor Imaging (BSD-DTI) is a novel approach to imaging of the diffusion tensor. By means of the method a spatial distribution of the b matrices is determined and subsequently incorporated into diffusion tensor calculation. This paper presents experimental verification of the method. Statistical analysis of the results shows significant variation of the b matrix components through the slices and spatial b matrix maps depict inhomogeneity of the gradient distribution for MRI scanners equipped with wide bore magnets. The accuracy of the diffusion tensor determination was improved by the factor of about 3 for the isotropic phantom using the Spin Echo Diffusion sequence.

On the ( $$1+3$$ 1 + 3 ) threading of spacetime with respect to an arbitrary timelike vector field

We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci tensor field of $(M, g)$ with respect to the threading frame field, in terms of the Ricci tensor field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.

Micro-machined super-miniature hot-film multi-array probe for field experiments with sub-Kolmogorov resolution

This is a report on the first successful experiments with a unique micro-hot-film multi-array probe. The main aim of this part of the project was to design, fabricate, implement and test in laboratory conditions a micro-hot-film multi-array probe for measurements in turbulent flows in field experiments with sub-Kolmogorov resolution. The key feature is that the probe is enabling to access, along with all three components of turbulent velocity fluctuations, also all nine components of the spatial velocity gradients tensor, including the option of obtaining the stream-wise velocity derivatives without employing the Taylor hypothesis, as well as velocity temporal derivatives. Hence the multi-array probe (typically five arrays) is built with each array consisting of four hot-film sensors, similarly to that of larger multi-array probe, based on arrays with four hot wires used in the atmospheric surface layer at Taylor micro-scale Reynolds number up to Reλ ≈ 104. This part of work relates to Reynolds numbers Reλ < 500 and employs several laboratory flows including flow in a channel of rectangular cross section, in its bulk and in the boundary layers in the proximity of its smooth and rough walls, and in the confined jet of circular cross section.

Self-Correcting Magnetic Tracking in Dynamic Medical Environments

We propose a novel self-correcting tracking technique that can minimize the dynamic distortion during magnetic tracking. In our method, the magnetic field distortion is estimated through field expansion and then minimized in an iterative tracking procedure. Sparse sensor arrays are designed to measure the spatial gradient tensors of the magnetic field for our method. Numerical results demonstrate that the tracking error caused by field distortion could be reduced from several tens of millimeters to below 2 mm.