Tensor Form Research Articles

Censored Least Squares for Imputing Missing Values in PARAFAC Tensor Factorization.

Tensor factorization is a dimensionality reduction method applied to multidimensional arrays. These methods are useful for identifying patterns within a variety of biomedical datasets due to their ability to preserve the organizational structure of experiments and therefore aid in generating meaningful insights. However, missing data in the datasets being analyzed can impose challenges. Tensor factorization can be performed with some level of missing data and reconstruct a complete tensor. However, while tensor methods may impute these missing values, the choice of fitting algorithm may influence the fidelity of these imputations. Previous approaches, based on alternating least squares with prefilled values or direct optimization, suffer from introduced bias or slow computational performance. In this study, we propose that censored least squares can better handle missing values with data structured in tensor form. We ran censored least squares on four different biological datasets and compared its performance against alternating least squares with prefilled values and direct optimization. We used the error of imputation and the ability to infer masked values to benchmark their missing data performance. Censored least squares appeared best suited for the analysis of high-dimensional biological data by accuracy and convergence metrics across several studies.

Hyperon semileptonic decays in QCD sum rules

We investigate the hyperon semileptonic decays within the framework of QCD sum rules. The flavor SU(3) symmetry breaking effects are analyzed via the relevant form factors and corresponding branching fractions. Employing the z-series parameterization to capture the q2 dependence of form factors, we calculate the hyperon semileptonic decay rates and confront them with the recent experimental measurements. Moreover, we calculate as well the non-standard tensor form factors, which involve certain new physics beyond the standard model.

Conceptual Study on Car Acceleration Strategies to Minimize Travel Time, Fuel Consumption, and CO2-CO Emissions

A conceptual study was performed on intelligent driving acceleration strategies for vehicles equipped with internal combustion engines. Two archetypal acceleration scenarios of highway driving and urban driving were prescribed. Three trajectories were considered for each scenario. They involved (a) nearly constant acceleration, (b) fast acceleration first and slow acceleration later, and (c) slow acceleration first and fast acceleration later. The selected vehicle was a generic European small–medium passenger car. Engine inlet pressure and ignition time were optimized along each trajectory to minimize fuel consumption, CO, and CO2 emissions, and travel time. The optimization process involved a methodological approach based on the higher-order singular value decomposition of the tensor form of the engine model. The optimized trajectories were analyzed and compared among themselves. Conceptual acceleration design guidelines for intelligent driving were provided that could be of interest when integrating vehicle/engine performance into the surrounding traffic flow.

A Mathematical Analysis and Simulation of the F-L Effect in Two-Layered Blood Flow through the Capillaries Remote from the Heart and Proximate to Human Tissue

In this paper, we have provided a mathematical analysis of an empirical result, namely, the Fahraeus–Lindqvist effect, a phenomenon that occurs in capillary tubes with a diameter lower than 0.3 mm. According to this effect, in capillary tubes under 0.3 mm in diameter, the apparent viscosity of blood decreases as the diameter of the tube decreases, making flow possible in these vessels. A two-phase blood flow mathematical model for human capillaries has been presented here. According to Haynes’ theory, blood is separated into two layers when it flows from the capillary. It is assumed that the first layer is plasma, and the second layer is the core layer. The plasma layer flows near the wall of the capillary, and the core layer flows along the axis of the capillary. Further, the core layer is assumed to be a mixture of two phases: one is the plasma, and the other is that of RBCs. For mathematical modeling purposes, a curvilinear coordinate system has been adopted, with physical quantities used in tensorial form. Derived equations are solved to find the effective viscosity, which depends upon the radius of the capillary; that is, it reduces viscosity to make blood flow possible. A comparative study was conducted with the experimental result of this effect, and it was observed that the proposed two-phase blood flow model is much closer to the experimental data than the single-phase blood flow model, and both have the same trends. After validation of the model with the experimental result, this model was applied to human capillaries (diameter lower than 10 μm) to show the F-L effect, and the impact of various physiological quantities that are relevant to the flow of blood into human capillaries are also discussed here. The impact of hematocrit on various parameters has been demonstrated explicitly.

Molecular Structure Refinement Based on Residual Dipolar Couplings: A Comparison of the Molecular Rotational-Sampling Method with the Alignment-Tensor Approach.

In NMR experiments, residual dipolar couplings (RDCs) in a molecule can be measured by averaging the dipolar couplings (DCs) over the rotational motion of a molecule in an environment that induces a slight anisotropic orientation distribution of the molecule. Since the shape of the anisotropic distribution cannot be measured, it is standard practice to use a particular orientation distribution of the molecule with respect to the magnetic field, in the form of a so-called alignment tensor (AT), to calculate RDC-values for the molecule. Since the same alignment tensor is commonly used to calculate the different RDCs of a molecule, this approach rests on the assumption that the rotational motion of the molecule is decoupled from its internal motions and that the molecule is rigid. The validity of these two assumptions is investigated for a small, simple molecule, using a relatively rigid atomic interaction function or force field and a more flexible one. By simulating the molecule using an orientation-biasing force an anisotropic rotational distribution can be generated, for which RDCs can be obtained. Using these RDCs as target RDCs when applying one of the two approaches of structure refinement based on RDCs, it can be investigated how well the target RDCs are approximated in the RDC restraining and whether the corresponding nonuniform orientation distribution is reproduced. For the relatively rigid version of the molecule, the AT approach reproduces the target RDC-values, although the nonuniform orientation distribution of the angle θab,H between the vector r⃗ab connecting two atoms a and b in the molecule and the vector representing the direction of the magnetic field H⃗ as generated in the orientation-biasing simulation cannot be reproduced in the AT RDC-restraining simulation. For the relatively flexible version of the molecule, the AT approach fails to reproduce both the target RDC values and the nonuniform orientation distribution. For biomolecules with flexible parts, the application of the AT approach is thus not recommended. Instead, a method based on sampling of the rotational and internal degrees of freedom of the molecule should be applied in molecular structure determination or refinement based on measured RDCs.

Onset of dynamic void coalescence in porous ductile solids

This paper investigates void growth and coalescence in porous ductile solids under dynamic loading condition. A physical definition for the onset of void coalescence in porous ductile solids under dynamic loading is proposed. The onset is deemed to occur when the third invariant of the tensorial form of the Hill–Mandel condition attains a zero value. The definition allows for systematic investigations on the effects of dimensionless stress rate κ and stress state, defined by the stress triaxiality T and Lode parameter L, on the onset of void coalescence via micromechanical analyses. The analyses reveal that the critical macroscopic effective strain for the onset of void coalescence displays an increasing–decreasing transition trend as the dimensionless stress rate increases, for all levels of T and L considered. The macroscopic effective strain at the transition is identified as the “ductile–brittle” transition strain. The dimensionless stress rate at which the transition strain occurs is found to be relatively constant. A mapping in the κ−T space for L=−1, representative of a generalized uniaxial tension typical in spall fracture experiments, is established which depicts regions where coalescence and non-coalescence can take place, as well as the ductile–brittle regions demarcated by a ductile–brittle transition curve. The results also show that the critical void volume fraction and macroscopic effective strain at the onset of void coalescence are insensitive to inertia at high stress triaxialities at L=−1.

B→D* and Bs→Ds* vector, axial-vector and tensor form factors for the full q2 range from lattice QCD

We compute the complete set of Standard Model (SM) and tensor B→D*ℓν¯ and Bs→Ds*ℓν¯ semileptonic form factors across the full kinematic range of the decays using second generation MILC nf=2+1+1 highly improved staggered quark (HISQ) gluon field configurations and HISQ valence quarks, with the method. Lattice spacings range from 0.09 to 0.044 fm with pion masses from ≈300 MeV down to the physical value and heavy quark masses ranging between ≈1.5mc and 4.1mc≈0.9mb; currents are normalized nonperturbatively. Using the recent untagged B→D*ℓν¯ℓ data from Belle and Bs→Ds*μν¯μ from LHCb together with our form factors, we determine a model independent value of Vcb=39.03(56)exp(67)latt×10−3, in agreement with previous exclusive determinations and in tension with the most recent inclusive result at the level of 3.6σ. We also observe a ≈1σ tension between the shape of the differential decay rates computed using our form factors and those measured by Belle. We compute a purely theoretical Standard Model value for the ratio of semitauonic and semimuonic decay rates, R(D*)=0.273(15), which we find to be closer to the recent Belle measurement and heavy flavor averaging group average than theory predictions using fits to experimental differential rate data for B→D*ℓν¯ℓ. Determining Vcb from our form factors and the experimental total rate for B→D*ℓν also gives a value in agreement with inclusive results. We also compute the longitudinal polarization fraction for the semitauonic mode, FLD*=0.395(24), which is in tension at the level of 2.2σ with the recent Belle measurement. Our calculation combines B→D* and Bs→Ds* lattice results in a simultaneous chiral continuum extrapolation, maintaining correlations between both modes. We then give results for both B→D* and Bs→Ds*, with the Bs→Ds* results superseding our previous lattice computation. We also give the chiral perturbation theory needed to analyze the tensor form factors. Published by the American Physical Society 2024

Energy-momentum tensor of the dilute (3+1)D glasma

We present a succinct formulation of the energy-momentum tensor of the glasma characterizing the initial color fields in relativistic heavy-ion collisions in the color glass condensate effective theory. We derive concise expressions for the (3+1)D dynamical evolution of symmetric nuclear collisions in the weak field approximation employing a generalized McLerran-Venugopalan model with nontrivial longitudinal correlations. Utilizing Monte Carlo integration, we calculate in unprecedented detail non-trivial rapidity profiles of early time observables at RHIC and LHC energies, including transverse energy densities and eccentricities. For our setup with broken boost invariance, we carefully discuss the placement of the origin of the Milne frame and interpret the components of the energy-momentum tensor. We find longitudinal flow that deviates from standard Bjorken flow in the (3+1)D case and provide a geometric interpretation of this effect. Furthermore, we observe a universal shape in the flanks of the rapidity profiles regardless of collision energy and predict that limiting fragmentation should also hold at LHC energies. Published by the American Physical Society 2024

A micromechanical scheme with nonlinear concentration functions by physics-guided neural network

The mechanical behavior of heterogeneous materials has been reported to be significantly influenced by the nonlinear properties of both the matrix and interface. However, the micromechanical homogenization methods for predicting the effective properties are challenged in nonlinear problems due to the difficulties in solving the analytical form of the concentration tensors. In this study, we develop a nonlinear micromechanical scheme for heterogeneous materials with complex interfacial behaviors, where the key component, namely nonlinear concentration functions, is determined by the devised physics-guided neural network. In particular, the nonlinear Mori–Tanaka method (NMT) implemented within this new micromechanical scheme yields accurate solutions to axisymmetric nonlinear homogenization problems considering the effects of finite deformation, loading conditions, volume fraction, etc. Furthermore, the NMT is equivalent to the linear Mori–Tanaka method in the condition of the small deformation. Notably, this neural-network-based micromechanical scheme shows good generalization for different types of nonlinear interfaces, while the corresponding approach for generating training data via the finite element method (FEM) is cost-effective. This theoretical framework introduces a novel approach to nonlinear physical modeling, namely, not by the direct regression from the dataset but by deeply embedding neural networks in physical laws.

Four-gluon vertex from the Curci-Ferrari model at one-loop order

We compute the four-gluon vertex from the Curci-Ferrari model at one-loop order for a collinear configuration. Our results display a good agreement with the first lattice data for this vertex, released very recently []. A noteworthy novelty of our work is that we can provide analytical expressions for the four-gluon vertex in collinear configurations, together with a renormalization scheme that allows us to perform reliable perturbative computations even in the infrared regime. We observe an infrared suppression in the form factor associated with the tree-level four-gluon tensor with a possible zero crossing in the deep infrared which demands new lattice investigations to be confirmed. Moreover, we report an infrared divergence in the completely symmetric tensor form factor due to the ghost-loop contributions. These results come as predictions, since previous two-point correlations fix all the available parameters of the model, up to an overall constant factor. Published by the American Physical Society 2024

Analysis of the rotationally-resolved 3.3 µm region of C2H4 in natural isotopic abundance

Ethylene (or ethene) is a natural gas compound present in our atmosphere but also in the atmospheres of outer solar system planets such as Jupiter, Saturn, Uranus, Neptune and Saturn's moon Titan. However, a better knowledge of spectroscopic parameters is needed for this molecule in order to detect its presence and/or to derive its accurate concentration. We present here a detailed analysis and modeling of the strongly absorbing ν9 and ν11 fundamental bands in the 3.3 µm region. Due to the complexity of the observed spectrum, we have built a polyad scheme taking into account some fundamental bands previously analyzed thanks to the tensorial formalism developed in Dijon for asymmetric-top molecules. A four polyad system has been settled and the last P3 polyad contains four vibrational states: ν9, ν11, ν2+ν12 and 2ν10+ν12. A first frequency analysis has been performed, leading to 3328 assignments with a root mean square of 5.927 × 10−3 cm−1 and a standard deviation of 1.965 using 88 adjusted parameters, and we have also adjusted line intensities of 2190 lines by adjusting 14 parameters with an overall Root-Mean-Square (RMS) deviation of about 3.77 %. A new linelist of calculated lines in the 2900−3300 cm−1 region has been added to the ECaSDa and we plan to integrate it to the HITRAN database.

Measurement and modeling of strain waves in germanium induced by ultrafast laser pulses

Transient reflectivity measurements are used to probe the strain waves induced by ultrashort laser pulses in bulk [100] germanium. The measurement signals are compared to purely analytical model functions based on the known material parameters for germanium. The modeling includes (i) a derivation of analytical solutions to the wave equation for strain waves coupled to the diffusion equation for heat and charge carriers and (ii) an expression for the impact on reflection coefficients that are caused by perturbations to the dielectric function but extended to cover a non-isotropic, uniaxial dielectric tensorial form. The model is held up against transient reflectivity measurements with an s- and a p-polarized probe and with a probe wavelength in the range of 502–710 nm. Excellent agreement is found when comparing the oscillatory shape of the measurement signals to the models. As for the magnitude of the oscillations, the models reproduce the overall trends of the experiment when using the previously published values for the elasto-optical tensor measured under static strain.

Nonlinear susceptibilities for weakly turbulent magnetized plasma: Electromagnetic formalism

This is a companion paper to the previous work [P. H. Yoon, Phys. Plasmas 31, 032309 (2024)] in which the nonlinear susceptibilities of weakly turbulent magnetized plasma are derived under a simplifying assumption of electrostatic interaction. The present paper extends the analysis to a general situation of electromagnetic interaction. The main novelty of the previous and present papers is that by employing the Bessel function addition theorem, the mathematical definitions for the susceptibilities are substantially simplified, a procedure that has not been discussed in the existing literature. In the present paper, a full set of Maxwell’s equations are considered in conjunction with the nonlinear Vlasov equation, which is solved by a perturbative method. The result is a fully general nonlinear susceptibility, given in tensorial form, which is applicable for weakly turbulent magnetized plasmas.

Absorptive nature of scattering coefficients in stress-energy tensor formalism for room acoustics.

In the stress-energy tensor formalism, the symmetry between absorption and scattering coefficients, as proven by measurements combined with simulations, is counterintuitive. By introducing the wall admittance, we show that the scattering coefficient is partly created by the real part of the wall admittance combined with the active intensity, that is, is partly due to absorption. However, for curved surfaces or finite source distances, it also depends on the imaginary part of the wall admittance in combination with the reactive intensity, which confers its genuine scattering properties inversely proportional to the distances to the sources. Thus, for plane waves impinging on plane boundaries, or purely real admittances, scattering reduces to absorption.

Continuum damage mechanics-based fatigue life prediction of L-PBF Ti-6Al-4V

The fatigue behavior and life of additively manufactured Ti-6Al-4V are very dependent on its heterogeneous microstructure, which in turn results from its fabrication process. In this paper, a fatigue damage model based on continuum damage mechanics was proposed to predict the high-cycle and very-high-cycle fatigue life at various stress ratios of Ti-6Al-4V fabricated by laser powder bed fusion (L-PBF). Model predictions were found to be in good agreement with experimental results for the different stress ratios considered. The model was found to be sufficiently robust to predict the fatigue life of other titanium alloys over a range of mean stress. The primary novelty is the incorporation of a tensorial formulation for anisotropic fatigue properties and the nonlinear effect of mean stress, which extended the model applicability to a wide range of stress ratio from -1 to 0.5. The results provide new insight into mean stress effects on fatigue of L-PBF Ti-6Al-4V up to the very-high-cycle fatigue regime.

Dalton's law of partial optical thermodynamic pressures in highly multimoded nonlinear photonic systems.

We show that in highly multimoded nonlinear photonic systems, the optical thermodynamic pressures emerging from different species of the optical field obey Dalton's law of partial pressures. In multimode settings, the optical thermodynamic pressure is defined as the conjugate to the extensive variable associated with the system's total number of modes and is directly related to the actual electrodynamic radiation forces exerted at the physical boundaries of the system. Here, we extend this notion to photonic configuration supporting different species of the optical field. Under thermal equilibrium conditions, we formally derive an equation that establishes a direct link between the partial thermodynamic pressures and the electrodynamic radiation pressures exerted by each polarization species. Our theoretical framework provides a straightforward approach for quantifying the total radiation pressures through the system's thermodynamic variables without invoking the Maxwell stress tensor formalism. In essence, we show that the total electrodynamic pressure in such arrangements can be obtained in an effortless manner from initial excitation conditions, thus avoiding time-consuming simulations of the utterly complex multimode dynamics. To illustrate the validity of our results, we carry out numerical simulations in multimoded nonlinear optical structures supporting two polarization species and demonstrate excellent agreement with the Maxwell stress tensor method.

Sedimentation of spheroids in Newtonian fluids with spatially varying viscosity

This paper examines the rigid body motion of a spheroid sedimenting in a Newtonian fluid with a spatially varying viscosity field. The fluid is at zero Reynolds number, and the viscosity varies linearly in space in an arbitrary direction with respect to the external force. First, we obtain the correction to the spheroid's rigid body motion in the limit of small viscosity gradients, using a perturbation expansion combined with the reciprocal theorem. Next, we determine the general form of the particle's mobility tensor relating its rigid body motion to an external force and torque. The viscosity gradient does not alter the force/translation and torque/rotation relationships, but introduces new force/rotation and torque/translation couplings that are determined for a wide range of particle aspect ratios. Finally, we discuss results for the spheroid's rotation and centre-of-mass trajectory during sedimentation. A steady orientation arises at long time whose value depends on the viscosity gradient direction and particle shape. These results are significantly different than when no viscosity gradient is present, where the particle stays at its initial orientation for all times. We summarize the observations for prolate and oblate spheroids for different viscosity gradient directions and provide plots for the orientation and centre-of-mass trajectory versus time. We also provide guidelines to extend the analysis when the viscosity gradient exhibits a more complicated spatial behaviour.

Conharmonically flat and conharmonically symmetric warped product manifolds

This article presents characterizations of warped product manifolds based on the flatness and symmetry of the conharmonic curvature tensor. It is proved that when a warped product manifold is conharmonically flat, both the base and fiber manifolds exhibit constant sectional curvatures. In addition, the specific forms of the conharmonic curvature tensor are derived for both the base and fiber manifolds. It is demonstrated that in a conharmonically symmetric warped product manifold, the fiber manifold has a constant sectional curvature, while the base manifold is both Cartan-symmetric and conharmonically symmetric. In this scenario, the form of the conharmonic curvature tensor on the fiber manifold is determined. We characterize the generalized Robertson–Walker (GRW) space-time through the flatness and the symmetry of the conharmonic curvature tensor. It is shown that a conharmonically flat (symmetric) GRW space-time is a perfect fluid. Finally, a conharmonically flat standard static space-time is investigated.

Tensor-Based Sparse Representation for Hyperspectral Image Reconstruction Using RGB Inputs

Hyperspectral image (HSI) reconstruction from RGB input has drawn much attention recently and plays a crucial role in further vision tasks. However, current sparse coding algorithms often take each single pixel as the basic processing unit during the reconstruction process, which ignores the strong similarity and relation between adjacent pixels within an image or scene, leading to an inadequate learning of spectral and spatial features in the target hyperspectral domain. In this paper, a novel tensor-based sparse coding method is proposed to integrate both spectral and spatial information represented in tensor forms, which is capable of taking all the neighboring pixels into account during the spectral super-resolution (SSR) process without breaking the semantic structures, thus improving the accuracy of the final results. Specifically, the proposed method recovers the unknown HSI signals using sparse coding on the learned dictionary pairs. Firstly, the spatial information of pixels is used to constrain the sparse reconstruction process, which effectively improves the spectral reconstruction accuracy of pixels. In addition, the traditional two-dimensional dictionary learning is further extended to the tensor domain, by which the structure of inputs can be processed in a more flexible way, thus enhancing the spatial contextual relations. To this end, a rudimentary HSI estimation acquired in the sparse reconstruction stage is further enhanced by introducing the regression method, aiming to eliminate the spectral distortion to some extent. Abundant experiments are conducted on two public datasets, indicating the considerable availability of the proposed framework.

Symmetry analysis involving meson mixing for charmonium decay

For charmonium’s decaying to the final states involving merely light quarks, in light of SU(3) flavor symmetry, the effective interaction Hamiltonian in tensor form is obtained by virtue of group representation theory. The strong and electromagnetic breaking effects are treated as a spurion octet so that the flavor singlet principle can be utilized as the criterion to determine the form of effective Hamiltonian for all charmonium two body decays. Moreover, a synthetic nonet is introduced to include both octet and singlet representations for meson description, and resorting to the mixing angle, the pure octet and singlet states are combined into the observable pseudoscalar and vector particles, so that the empirically effective Hamiltonian can be obtained in a concise way. As an application, by virtue of this scenario the relative phase between the strong and electromagnetic amplitudes is studied for vector-pseudoscalar meson final state. In data analysis of samples taken in an e+e− collider, the details of experimental effects, such as energy spread and initial state radiative correction, are taken into consideration in order to make full use of experimental information and acquire accurate and delicate results. Published by the American Physical Society 2024