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The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all $$R^{K}_{{\varvec{0}}}$$ -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an $$ER^{K}$$ -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature.

In the preset exploration, a numerical strategy based upon the moving Kriging meshfree formulations is proposed to analyze small scale-dependent nonlinear oscillations of random checkerboard reinforced cylindrical microshells. To accomplish this aim, the modified couple stress continuum mechanics is implemented in the third-order shear flexible shell theory. The microshells are made of composites containing graphene nanofillers dispersed in a random checkerboard pattern. The associated material characteristics are captured via a probabilistic-based micromechanical scheme together with the Monte-Carlo simulation. Afterwards, proper meshfree functions are implemented to enforce the essential boundary conditions at the considered nodding system accurately. It is revealed that that by considering the effects of rotation gradient tensor, the frequency ratio associated with a given maximum shell deflection reduces, while the value of linear frequency gets larger. This anticipation is related to stiffening characters of this microstructural gradient tensor. Moreover, it is demonstrated that an increment in the value of the nanofiller volume fraction or its aspect ratio leads to shift the peak of the frequency ratio to a lower value of the shell length to radius ratio.

The Donaldson-Fujiki K\"ahler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of K\"ahler metrics as a moment map, can be lifted canonically to a hyperk\"ahler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperk\"ahler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to $K$-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the ``Higgs tensor''. We also discuss some aspects of the analogy with Higgs bundles., Comment: 43 pages. Accepted version. Upgraded the previous results to consider uniform toric K-stability

We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for 4009 nonlinear Schrodinger equations in dimension two, arXiv:1910.08492 ), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrodinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).

We study the semiclassical Einstein field equations with a Klein-Gordon field in ultrastatic and static spacetimes. In both cases, the equations for the spacetime metric become constraint equations. In the ultrastatic case, the Hadamard singular structure can be characterised explicitly, which allows one to in principle give initial data for the Wightman function that has correct distributional singularities, such that the expectation value of the renormalised stress-energy tensor of the solution can be defined, and that hence the semiclassical Einstein equations make sense. Assuming a "positive energy" condition for the Klein-Gordon operator, we characterise the states for which, if the constraints hold for initial data, they hold everywhere in spacetime. These turn out to be time-translation invariant states. The static case is analysed by conformal techniques, effectively reducing the problem to an ultrastatic one., The previous version has been reorganised and expanded and minor errors have been fixed. 25 pages

In the context of a purely mechanical development for materials that possess some degree of elasticity, two-factor and three-factor multiplicative decompositions of the deformation gradient, as well as related strain measures, are discussed in detail. Different factors in the decompositions (and their polar subfactors) have different degrees of rotational non-uniqueness. Moreover, different deformation measures generally behave differently under superposed rigid motions, which has important implications for the roles that they are allowed to play in constitutive equations. A certain unique right stretch tensor emerges, which yields strain measures suitable for describing anisotropic elastic responses of solids that have evolving stress-free local configurations.

In the present paper, we propose two new methods for tensor completion of third-order tensors. The proposed methods consist in minimizing the average rank of the underlying tensor using its approximate function, namely the tensor nuclear norm. The recovered data will be obtained by combining the minimization process with the total variation regularization technique. We will adopt the alternating direction method of multipliers, using the tensor T-product, to solve the main optimization problems associated with the two proposed algorithms. In the last section, we present some numerical experiments and comparisons with the most known image video completion methods.

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states.

The inter-particle friction is mainly related to the roughness of particles, which plays an important role in both macroscopic and microscopic mechanical behaviors of granular materials. Under the real three-dimensional (3D) loads, the multiscale effects of inter-particle friction could be more complex. In this study, a series of true triaxial tests are conducted based on discrete element method (DEM). We identify that the inter-particle friction is key to triggering the disaccord of Lode angles between stress tensor and incremental strain tensor (also fabric tensor) on π-plane under 3D non-axisymmetric loading paths. A kind of function form is proposed to describe this disaccord, by incorporating the inter-particle friction coefficient μ and intermediate principal stress ratio b. We further observe that the evolution of strong contact network coincides with the overall stress tensor in the Lode coordinator system, irrespective of μ. The disaccord behaviors of overall contact fabric and the macroscopic stress mainly originate from the weak subset of contacts, and then, the inter-particle friction should play an important role on the rearrangements of weak contact network and finally lead to the disaccord behaviors at the macroscopic scale.

Traversable wormhole spacetimes are constructed in Palatini version of modified f(R) gravity theory. We impose that the matter supporting these spacetimes satisfies the null energy condition (NEC) and the weak energy condition (WEC). The gravitational field equations in Palatini approach are totally different to those field equations obtained in general relativity and f(R) metric approach. In this framework, by considering $$f(R)= R - \beta R^n$$ which is a given f(R) model that has consistent results with observations, the field equations have been simplified. To explicitly examine the NEC and the WEC, we take three different shape functions $$b(r)=r_0^2/r$$ , $$b(r)=\sqrt{r_0r}$$ , and $$b(r)=r_0^3/r^2$$ . Then, the existence of traversable wormholes in Palatini f(R) gravity is investigated in the absence of exotic matter. It is clearly demonstrated that the matter stress-energy tensor satisfies the NEC and the WEC.

A method is proposed to identify the fully anisotropic elasticity tensor by applying the impulse excitation technique. A specially designed batch of several differently oriented bar-shaped specimens with rectangular cross section is analyzed with respect to the eigenfrequencies of their fundamental flexural and torsional modes. Estimations based on the equations for the calculation of the isotropic Young’s modulus and the shear modulus from the ASTM standard allow a first approximation of the elasticity tensor from a selected subset of the measured eigenfrequencies. Subsequently, a more precise determination of the elasticity tensor is achieved by a numerical modal analysis using the finite element method. In this course, a Newton–Raphson optimization method is applied to solve the inverse problem. The proposed approach is demonstrated on a batch of specimen fabricated from the nickel-base alloy IN718 by selective laser melting.

To locate all C-eigenvalues of a piezoelectric-type tensor, a new C-eigenvalue inclusion interval is constructed. It is proved that the new interval is tighter than that proposed by Wang et al. (Appl Math Lett 100:106035, 2020). Numerical examples show the effectiveness of the new result.

The topic of direction-of-arrival (DOA) estimation has attracted extensive attention in wireless communications, radars, sonars, etc. Compared to the traditional scalar sensor, electromagnetic vector sensor (EMVS) is attractive since it provides two-dimensional (2D) DOA estimation and additional polarization information of the incoming signals. However, existing algorithms are only suitable for Gaussian white noise scenario. In this paper, we investigate into DOA estimation using EMVS array with spatially colored noise, and a parallel factor (PARAFAC) estimator is proposed. Unlike the conventional direct PARAFAC algorithm, the covariance tensor-based PARAFAC model is considered. For fast PARAFAC decomposition purpose, the fourth-order covariance PARAFAC tensor is rearranged into a third-order PARAFAC tensor, so that the existing COMFAC algorithm is available and thus the factor matrices are estimated. Thereafter, the elevation angle estimation is accomplished via least squares (LS) technique, and the azimuth angles are estimated via vector cross-product. Besides, the polarization status of the source signal can be estimated via LS approach, which may be helpful to identify weak signals. Our estimator is flexible since it can be easily extended to nonuniform array and spatially colored noise scenario. Moreover, it offers better estimation performance than the traditional PARAFAC algorithm in the presence of colored noise. Detailed analyses concerning identifiability, complexity as well as Cramer–Rao bound (CRB) are provided. To show the effectiveness of the proposed estimator, numerical simulations have been designed.

Data stream classification is one of the most vital areas of contemporary machine learning, as many real-life problems generate data continuously and in large volumes. However, most of research in this area focuses on vector-based representations, which are unsuitable for capturing properties of more complex multi-dimensional structures, such as images and video sequences. In this paper, we propose a novel methodology for learning adaptive decision trees from data streams of tensors. We introduce Chordal Kernel Decision Tree for continual learning from tensor data streams. In order to maintain the tensor characteristics, we propose to train and update classifiers in the kernel space designed to work with tensor representation. We use chordal distance to compute similarities between tensors and then apply it as a new feature space in which decision trees are trained. This allows for a direct decision tree induction on tensors. In order to accommodate the streaming and drifting nature of data, we propose a concept drift detection scheme based on tensor representation. It allows us to reconstruct the kernel feature space every time when change is detected. The proposed approach allows for fast and efficient induction of decision trees on streaming data with tensor representation. Experimental study, conducted on 4 real-world and 52 artificial large-scale tensor data streams, shows that using the native tensor feature space leads to more accurate classification than outperforms the vectorized representations.

This article investigates reduced-order models (ROMs) based on proper orthogonal decomposition for efficient computation of the homogenized diffusion properties of saturated porous media. The homogenized tensor, whose classical expression may be obtained from periodic homogenization techniques, is computed by solving a local problem on an elementary cell that includes one or several circular solid inclusions. The cost of the repeated resolution of this problem by the finite element method for different inclusions radii may be important, and classical model order reduction methods based on the computation of a spatial basis cannot be applied directly. The method proposed in this work to cope with the variability of circular inclusions relies on the introduction of a transformation from a reference domain to the physical domain that admits an exact affine decomposition in the three-dimensional cases, allowing to split the problem into an offline learning phase and an online evaluation phase which does not depend on the number of degrees of freedom of the original full-order solution. Approximate affine decomposition for two-dimensional cases is also provided with an explicit estimation of the truncation error. The efficiency of the proposed algorithm in terms of accuracy and of computing time is evaluated first for 2D and 3D isotropic and anisotropic elementary cells with a single inclusion, and second for a 2D anisotropic cell with multiple inclusions. Furthermore the ROM is used to estimate in quasi-real time the homogenized diffusion tensor for a given probability distribution of the geometry parameters.

The existing methods for performing the super-resolution of the three-dimensional images are mainly based on the simple learning algorithms with the low computational powers and the complex deep learning neural network-based learning algorithms with the high computational powers. However, these methods are based on the prior knowledge of the images and require a large database of the pairs of the low-resolution images and the corresponding high-resolution images. To address this difficulty, this paper proposes a method based on the joint generalized singular value decomposition and tensor decomposition for performing the super-resolution. Here, it is not required to know the prior knowledge of the pairs of the low-resolution images and the corresponding high-resolution images. First, an image is represented as a tensor. Compared to the three-dimensional singular spectrum analysis, the spatial structure of the local adjacent pixels of the image is retained. Second, both the generalized singular value decomposition and the Tucker decomposition are applied to the tensor to obtain two low-resolution tensors. It is worth noting that the correlation between these two low-resolution tensors is preserved. Also, these two decompositions achieve the exact perfect reconstruction. Finally, the high-resolution image is reconstructed. Compared to the de-Hankelization of the three-dimensional singular spectrum analysis, the required computational complexity of the reconstruction of our proposed method is much lower. The computer numerical simulation results show that our proposed method achieves a higher peak signal-to-noise ratio than the existing methods.

This paper presents a system of cyclic tensor algebra for operations involving fourth-order tensors. The advantages are that the system is objectively and consistently defined in three ways that each fall into one of three universal classes. Operators within a given class are called conjugate operators such that many familiar and fundamental identities of scalars and second-order tensors are maintained in fourth order; this provides greater insight along with anthropological and pedagogical advantages over current systems, while also revealing new identities and solutions. The relationship between operators of a different class is such that a property of cyclic symmetry arises whereby mixed-class product operators can be cycled around without invalidating an equation. In defining this system, we have considered the following: preservation from identities in zeroth- (scalar) and second-order tensor identities to fourth-order tensor identities; possible permutations of definitions and subsequent logical restrictions; the visual notational consistency throughout the system; and maintenance to legacy definitions and operator symbols. Additionally, we present many new and useful algebraic identities and provide a comparison to some selected contemporary systems used in the literature. We also provide, to complete at least a basic exposition of our proposed system, a set of identities for matrix-equivalent operations, which facilitate programming for numerical computing. This article is designed to be used as a reference work for anyone choosing to adopt this system of tensor operations in continuum mechanics theory involving fourth-order tensors.

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, $$3+11$$ 3 + 11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and $$1+6$$ 1 + 6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

We develop an algorithmic framework for contracting tensor networks and demonstrate its power by classically simulating quantum computation of sizes previously deemed out of reach. Our main contribution, index slicing, is a method that efficiently parallelizes the contraction by breaking it down into much smaller and identically structured subtasks, which can then be executed in parallel without dependencies. We benchmark our algorithm on a class of random quantum circuits, achieving greater than 105 times acceleration over the original estimate of the simulation cost. We then demonstrate applications of the simulation framework for aiding the development of quantum algorithms and quantum error correction. As tensor networks are widely used in computational science, our simulation framework may find further applications. An efficient method for parallelizing the contraction of tensor networks pushes the boundaries for the classical simulation of quantum computation, and aids the development of quantum algorithms and hardware.

In recent years, tensor data appear more frequently in machine learning. Tucker decomposition is a powerful tool for processing tensor data. However, from the perspective of dimensionality reduction, Tucker decomposition without any regularization is only a tensor version of principal component analysis (PCA) that learning a linear subspace to minimize the distance between the data and their projections on the subspace. However, many high-dimensional tensor data usually reside in low-dimensional submanifolds, rather than low-dimensional linear subspaces. It is necessary to fill the gap between submanifolds and linear subspaces to achieve superior performance. In this paper, we proposed a new dimensionality reduction algorithm of tensor data based on manifold regularized Tucker decomposition (called MRTD for short), in which manifold regularization is in addition to Tucker decomposition. Furthermore, unlike many other similar algorithms that ignore the matrices of Tucker decomposition by absorbing them into a whole matrix, in this paper, we proposed an iterative solution to MRTD that the matrices are calculated based on each other iteratively. Matrices in Tucker decomposition represent different dimensionality reduction operations for each dimension of tensor. Therefore, MRTD is a dimensionality reduction algorithm specially designed for tensor data, not for vector data. The experimental results of MRTD and 5 other related state-of-the-art algorithms on 6 commonly-used real-world datasets are presented, showing the better performance of MRTD.

This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for $$\beta < 0.5$$ ) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.

Halide double perovskite Cs2AgBiBr6 shows promising potential applications in next-generation photovoltaic devices. The strain engineering strategy has been proven as a reliable method to improve its performance. The fundamental physical tensor of elastic constants Cij has not been accurately measured yet, due to the limited size of samples. The aim of this work is to measure its full elastic tensor Cij using various techniques. We utilized nanoindentation and contact resonance atomic force microscopy (CR-AFM) techniques to measure the elastic modulus on an as-grown facet, which is along the [111] direction. Then, the inelastic X-ray scattering (IXS) approach and density functional theory were introduced to obtain the accurate full elastic constant Cij of single crystal Cs2AgBiBr6. The full elastic constants were measured as C11 = 44.52 GPa, C12 = 17.63 GPa and C44 = 8.56 GPa. The results for the modulus along the [111] direction reveal that nanoindentation and CR-AFM are sufficiently reliable for quantitative measurements of small single crystals Cs2AgBiBr6. The results provide guidelines for strain engineering in the photovoltaic applications using Cs2AgBiBr6 and for elastic property measurements in nanomaterials.

A variation model of the gradient anisotropic elasticity theory is constructed. Its distinctive feature is the fact that the density of potential energy, along with symmetric fourth- and sixth-rank stiffness tensors, also contains a nonsymmetric fifth-rank stiffness tensor. Accordingly, the stresses in it also depend on the curvatures and the couple stresses depend on distortions. The Euler equations are three fourth-order equilibrium equations. The spectrum of boundary-value problems is determined by six pairs of alternative boundary conditions at each nonsingular surface point. At each special point of the surface (belonging to surface edges), in the general case, additional conditions arise for continuity of the displacement vector and the meniscus force vector when crossing the surface through the edge. In order to reduce the number of the physical parameters requiring experimental determination, particular types of the fifth- and sixth-rank stiffness tensors are postulated. Along with the classical tensor of anisotropic moduli, it is proposed to introduce a first-rank stiffness tensor (a vector with a length dimension), with the help of which the fifth-rank tensor is constructed from the classical fourthrank tensor by means of tensor multiplication. The sixth-rank tensor is constructed as the tensor product of the classical fourth-rank tensor and two first-rank tensors.

Recently, a new method to simplify calculations in teleparallel theories has been put forward. In this article, this method is improved and used to analyze the gravitational energy–momentum tensor (density) of the Schwarzschild solution in a frame that is arbitrarily accelerated along the z-direction. It is shown that, for a special type of frame, one cannot make the gravitational energy density vanish along the observer’s worldline, regardless of the observer’s acceleration. The role played by the frame and its relation to the observers’ worldlines are investigated. It is shown that, for the aforementioned special frames, the results for the gravitational energy and angular momenta are consistent with what we would expect.

A description of inverse energy cascade (from small scale to large scale) in homogeneous isotropic turbulence is introduced by using an eigenvalue method. We show a special isotropic turbulence, in which the initial condition is constructed by reversing the velocity field in space, i.e., the time-reversed turbulence. It is shown that the product of eigenvalues of the rate-of-strain tensor can quantitatively describe the backward energy transfer process. This description is consistent to the velocity derivative skewness Sk. However, compared with Sk, it is easier to be obtained, and it is expected to be extended to anisotropic turbulence. Furthermore, this description also works for the resolved velocity field, which means that it can be used in engineering turbulent flows. The description presented here is desired to inspire future investigation for the modeling of the backward energy transfer process and lay the foundation for the accurate prediction of complex flows.

In this paper, we extend the Dashnic-Zusmanovic inclusion sets of matrices to the tensor case, which is proved to be always tighter than the Brauer-type inclusion sets of tensors introduced by Bu et al. (Linear Algebra Appl 512:234–248, 2017). As applications, we presented that, a DZ-type Z-tensor with nonnegative diagonal entries a strong M-tensor, and an even order DZ-type symmetric tensor with positive diagonal entries is positive definite.

In this paper, we first extend the concept of non-degenerate matrices to tensors and we then study the finiteness properties of the solution set of non-degenerate tensor complementarity problems. When the involving tensor in the tensor complementarity problem is a positive linear combination of rank-one symmetric tensors, we show that the solution set of the tensor complementarity problem is convex if the underlying tensor is positive semidefinite, and the tensor complementarity problem has the globally uniqueness solvable property if the underlying tensor is positive definite. Finally, we prove that a symmetric P tensor with an additional condition has the globally uniqueness solvable property.

This paper introduces a novel approach to computing logic programming semantics. First, a propositional Herbrand base is represented in a vector space and if-then rules in a program are encoded in a matrix. Then the least fixpoint of a definite logic program is computed by matrix-vector products with a non-linear operation. Second, disjunctive logic programs are represented in third-order tensors and their minimal models are computed by algebraic manipulation of tensors. Third, normal logic programs are represented by matrices and third-order tensors, and their stable models are computed. The result of this paper exploits a new connection between linear algebraic computation and symbolic computation, which has the potential to realize logical inference in huge scale of knowledge bases.

Exploiting the prior information is fundamental for image reconstruction in computational hyperspectral imaging (CHI). Existing methods usually unfold the 3D signal as a 1D vector and then handle the prior information among different dimensions in an indiscriminative manner, which inevitably ignores the high-dimensionality nature of the hyperspectral image (HSI) and thus results in poor reconstruction performance. In this paper, we propose a high-order tensor optimization based reconstruction method to boost the quality of CHI. Specifically, we first propose an adaptive dimension-discriminative low-rank tensor recovery (ADLTR) model to exploit the high-dimensionality prior of HSI faithfully. In the ADLTR model, we utilize the 3D tensors as the basic elements to fundamentally preserve the structure information in the spatial and spectral dimensions, introduce a dimension-discriminative low-rankness model to fully characterize the prior in the basic elements, and propose a weight estimation strategy by adaptively exploiting the diversity in each dimension. Then, we develop an optimization framework for the CHI reconstruction by integrating the structure prior in ADLTR with the system imaging principle, which is finally solved via the alternating minimization scheme. Extensive experiments on both synthetic and real data demonstrate that our method outperforms state-of-the-art methods.

Given two operator spaces E and F, injectivity of the canonical map from the $$\lambda $$ -tensor product $$E\bigotimes ^\lambda F$$ into the operator space injective tensor product $$E\bigotimes ^{\min } F$$ is investigated, where the $$\lambda $$ -tensor product is a generalization of the Haagerup, the operator space projective and the Schur tensor products. Properties related to the spectra of elements in $${\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ , when $${\mathcal {A}} $$ and $${\mathcal {B}}$$ are two $$\hbox {C}^*$$ -algebras, are explored. One of the two important questions that are discussed is regarding the spectrum of an elementary tensor $$a\otimes b$$ in terms of the spectra of $$a\in {\mathcal {A}}$$ and $$b\in {\mathcal {B}}$$ , and the other is about the possibility of determining elements of the tensor product from their spectra in the sense that two elements $$u,v\in {\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ are equal if and only if the nonzero spectral values of ux and vx coincide for every $$x\in {\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ .

This paper presents a novel, exact, semi-analytical solution for the quasi-static drained expansion of a cylindrical cavity in soft soils with fabric anisotropy and structure. The assumed constitutive model is the S-CLAY1S model, which is a Cam clay-type model that considers fabric anisotropy that evolves with plastic strains, structure and gradual degradation of bonding (destructuration) due to plastic straining. The solution involves the numerical integration of a system of eight first-order ordinary differential equations, three of them corresponding to the effective stresses in cylindrical coordinates, other three corresponding to the components of the fabric tensor and one corresponding to the amount of bonding and another corresponding to the specific volume. The solution is validated against finite element analyses. When destructuration is considered, the solution provides slightly lower values of the effective radial and mean stresses near the cavity wall. Besides, the specific volume is further reduced due to loss of bonding. Parametric analyses and discussion of the influence of soil overconsolidation, expansion of the cavity and initial amount of bonding are presented.

Since the first record of Raman spectra of graphite in 1970, the physical origin behind its Raman characteristic peaks (i.e., G, D and D' peaks) has been a focus of controversy. At present, it is generally believed that G peak corresponds to Raman active E 2g2 mode, while D and D' peaks are defect-induced ones. However, unequivocal experimental evidence for the phonon symmetries for these graphite Raman peaks is almost still in blank. Here, we clarify these important aspects using an angle resolved polarized Raman spectroscopy. It is found that the experimental Raman intensity of D and D' peaks shows a similar polarized angle dependence as that of G peak. Combined with Raman tensor analysis and double-resonant mechanism, the phonon symmetry of D' and D peak is further understood. Our work provides reliable experimental evidence and reasonable explanation for better understanding the phonon symmetry of graphite.

The Fourier matrix is fundamental in discrete Fourier transforms and fast Fourier transforms. We generalize the Fourier matrix, extend the concept of Fourier matrix to higher order Fourier tensor, present the spectrum of the Fourier tensors, and use the Fourier tensor to simplify the high order Fourier analysis.

The pure cross-anisotropy is understood as a special scaling of strain (or stress). The scaled tensor is used as an argument in the elastic stiffness (or compliance). Such anisotropy can be overlaid on the top of any elastic stiffness, in particular on one obtained from an elastic potential with its own stress-induced anisotropy. This superposition does not violate the Second Law. The method can be also applied to other functions like plastic potentials or yield surfaces, wherever some cross-anisotropy is desired. The pure cross-anisotropy is described by the sedimentation vector and at most two constants. Scaling with more than two purely anisotropic constants is shown impossible. The formulation was compared with experiments and alternative approaches. Static and dynamic calibration of the pure anisotropy is also discussed. Graphic representation of stiffness with the popular response envelopes requires some enhancement for anisotropy. Several examples are presented. All derivations and examples were accomplished using the algebra program Mathematica.

This communication provides initial information and understanding of the manner in which a newly developed theoretical mechanism (Soldatos in Int J Solids Struct 202:217–225, 2020) is applied in specific boundary value problems met in polar linear elasticity of fibrous composites and thus enables the determination of the spherical part of the couple-stress tensor. In this context, it tests the applicability of the implied mechanism/method in the case that a rectangular plate reinforced by a single family of unidirectional fibres is subjected to pure bending. The problem solution is obtained for either non-polar or polar material behaviour, where fibres are considered perfectly flexible or resistant in bending, respectively, and provides clear evidence of the correctness of the principal argument that underpins the proposed method. Namely, that the general rotation field of the plate deformation differs from the fibre rotation field. That newly discovered method enables an extra energy term that emerges in the strain energy function of the fibrous composite plate to relate with the spherical part of the couple-stress tensor outside conventional equilibrium conventions. It thus leads to the determination of the spherical part of the couple-stress and its distribution throughout the plate body in a complete and comprehensive manner.

The natural frequencies of structures submerged in a fluid are significantly influenced by fluid properties and adjacent structures, which can be expressed by added mass. Some finite element software can effectively determine the added mass matrix of arbitrary structures immersed in viscous fluids, provided the structure lengths are relatively long; however, the added mass specified in a second-order tensor cannot be used as is. Herein, we propose methods to calculate the coupled added mass as a single value for each structure using the MassInFlu code. When the x- and y-directional components of the coupled added mass are unequal, an effective added mass can be determined by rotating the structures or coordinate axes. Various examples considering different scenarios show that the coupled natural frequencies obtained using the coupled or effective added mass coincide with those obtained from fluid-structure interaction analyses.

At the present time, the classical theory of radiation of charged particles is a complete theory. It has wide applications in the generation of various radiation sources: synchrotron radiation, undulator radiation, and others. In this regard, questions of radiation of the intrinsic magnetic moment of the particles have not been completely investigated. A path to the construction of radiation fields of a point source is laid out within the formalism of the Hertz tensor. A derivation of potentials and field strengths of a point radiation source is given.

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m − 1)-homogeneous operator from l1 into lp (1 < p < ∞), and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, sufficient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.

We examine homogeneous solitons of the ambient obstruction flow and, in particular, prove that any compact ambient obstruction soliton with constant scalar curvature is trivial. Focusing on dimension 4, we show that any homogeneous gradient Bach soliton that is steady must be Bach flat, and that the only non-Bach-flat shrinking gradient solitons are product metrics on $$\mathbb {R}^2\times S^2$$ and $$\mathbb {R}^2 \times H^2$$ . We also construct a non-Bach-flat expanding homogeneous gradient Bach soliton. We also establish a number of results for solitons to the geometric flow by a general tensor q.

The majority of the data produced by human activities and modern cyber-physical systems involve complex relations among their features. Such relations can be often represented by means of tensors, which can be viewed as generalization of matrices and, as such, can be analyzed by using higher-order extensions of existing machine learning methods, such as clustering and co-clustering. Tensor co-clustering, in particular, has been proven useful in many applications, due to its ability of coping with n-modal data and sparsity. However, setting up a co-clustering algorithm properly requires the specification of the desired number of clusters for each mode as input parameters. This choice is already difficult in relatively easy settings, like flat clustering on data matrices, but on tensors it could be even more frustrating. To face this issue, we propose a new tensor co-clustering algorithm that does not require the number of desired co-clusters as input, as it optimizes an objective function based on a measure of association across discrete random variables (called Goodman and Kruskal’s $$\tau$$ τ ) that is not affected by their cardinality. We introduce different optimization schemes and show their theoretical and empirical convergence properties. Additionally, we show the effectiveness of our algorithm on both synthetic and real-world datasets, also in comparison with state-of-the-art co-clustering methods based on tensor factorization and latent block models.

For finite-strain plasticity with anisotropic yield functions and anisotropic hyperelasticity, we use the Kroner-Lee decomposition of the deformation gradient combined with a yield function written in terms of the Mandel stress. The source is here the right Cauchy-Green tensor provided by a FE discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential, which is compared with a classical backward-Euler method. The exact Jacobian of the second Piola-Kirchhoff stress is determined with respect to this source, consistent with the approximation. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function. The constitutive system is solved by a damped Newton-Raphson algorithm for the plastic multiplier and the elastic right Cauchy-Green tensor $$\varvec{C}_{e}$$ . To ensure power-consistency, we make use of the elastic Mandel stress construction. Two numerical examples exhibit the comparative effectiveness of the Algorithm for very large elastic and plastic deformations. The elasto-plastic pinched cylinder makes use of as few as 2 steps for the total radius displacement of 300 mm and only 25 steps are required for the cup drawing problem.

The cerebral atlas of diffusion tensor magnetic resonance image (DT-MRI, shorted as DTI) is one of the key issues in neuroimaging research. It is crucial for comparisons of neuronal structural integrity and connectivity across populations. Usually, the atlas is constructed by iteratively averaging the registered individual image. In tradition, the fuzzy group average image is easily generated in the initial stage, which is harmful to providing clear guidance for subsequent registration, to the performance of the final atlas. To solve this problem, an improved unbiased DTI atlas construction algorithm based on adaptive weights is proposed in this paper. The adaptive weighted strategy based on diffeomorphic deformable tensor registration is introduced. At the same time, the distance measure for tensors is used as a constraint condition, which ensures the unbiasedness of the atlas. Then, using 77 DTIs from the dataset in http://www.brain-development.org , three study-specific atlases, i.e. the constructed atlases of the proposed algorithm and two open-sourced algorithms (DTIAtlasBuilder and DTI-TK), are compared with two standardized atlases (IIT v. 4.1 and NTU-DSI-122-DTI). The performances of the atlases were evaluated in spatial normalization way with six region-based criteria (including Euclidean distances between diffusion tensors, Euclidean distances of the deviatoric tensors, standard deviation, overlaps of eigenvalue-eigenvector, cross-correlations and three sets angles of eigenvalue-eigenvector pairs between diffusion tensors) and three fiber-based criteria (including distances between fiber bundles, angles between fiber bundles and fiber property profile-based criteria). The experimental results showed that the overall performances of the study-specific atlases are better than those of the standardized atlases for specific datasets, and the comprehensive performance of the improved algorithm proposed in this paper is the best.

A real symmetric tensor $${\mathscr{A}} = ({a_{{i_1} \ldots {i_m}}}) \in {\mathbb{R}^{[m,n]}}$$ is copositive (resp., strictly copositive) if $${\mathscr{A}}\;{x^m} \geqslant 0$$ (resp., $${\mathscr{A}}\;{x^m} > 0$$ ) for any nonzero nonnegative vector x ∈ ℝn. By using the associated hypergraph of $${\mathscr{A}}$$ , we give necessary and sufficient conditions for the copositivity of $${\mathscr{A}}$$ . For a real symmetric tensor $${\mathscr{A}}$$ satisfying the associated negative hypergraph $${H_ - }({\mathscr{A}})$$ and associated positive hypergraph $${H_ + }({\mathscr{A}})$$ are edge disjoint subhypergraphs of a supertree or cored hypergraph, we derive criteria for the copositivity of $${\mathscr{A}}$$ . We also use copositive tensors to study the positivity of tensor systems.

A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $\mu\vdash d$ and a $\mu$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{\mu,\mathbb{R}}$ of $\mu$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_\mu$ the complexification of $X_{\mu,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ of the dual variety of $X_\mu$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ is the square of the $\mu$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $\mu$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ for tensors of format $2\times 2\times 2$., Comment: 21 pages, 5 figures

Pairwise rigid registration can be developed by comparing local geometry encoded by intrinsic second-order orientation tensors which allows to model the registration problem using the associated Riemannian geometry or related group structures. Everything starts by representing those tensor fields as multivariate normal models that permit us to manipulate Gaussians in two ways: using the Riemannian manifold elements, that can be embedded into matrix spaces with geometric structures, or through Lie group/algebra techniques. In this paper we discuss some points behind these approaches in the context of rigid registration problems. Firstly, they are not equivalent since, in general, there is no isometry linking them. Secondly, embedding methodologies are not invariant with respect to rigid motion. We discuss these points using two variants of the Iterative Closest Point that use the comparative tensor shape factor (CTSF) to match orientation tensors. We replace the CTSF to different criteria computed through geodesic distance and algebraic embeddings and compare the registration algorithms showing that the latter is more efficient for registration of point clouds.

For a hypergraph H, let B(H) be its incidence matrix. The signless Laplacian matrix $$Q(H)=B(H)B(H)^T$$ , whose (i, j)-element is exactly the number of edges containing vertices $$v_i$$ and $$v_j$$ for $$i\ne j$$ and (i, i)-element is exactly the degree of vertex $$v_i$$ . The incidence Q-tensor $${\mathcal {Q}}^*=B(H){\mathcal {I}}B(H)^T$$ , whose $$(i_1,i_2,\ldots ,i_k)$$ -entry of $${\mathcal {Q}}^*$$ is exactly equal to the number of edges e of H satisfying $$i_t\in e$$ for all $$t\in [k]$$ . Obviously, we can see more complete structural properties of H from $${\mathcal {Q}}^*$$ than Q(H). Up to now, few tensors whose entries are directly related to structural properties of the corresponding hypergraphs. In this regard, we believe that it deserve to study the structural properties of hypergraphs though this tensor. In this paper, we will study some upper bounds on the spectral radius of $${\mathcal {Q}}^*$$ by some parameters on hypergraphs.

In this paper, we have investigated the exact solutions of a perfect fluid Bianchi type $$-I$$ cosmological model in the context of $$f(R, T) = R + \lambda T$$ gravity, where R is Ricci scalar, and T is a trace of the energy-momentum tensor. We show that the generalized Friedman equation’s exact solution for the average scale factor a involves the hypergeometric function. This approach is novel, and it has not been previously carried out by other authors. Consequently, a set of three cosmological models, namely dust, radiation and dark energy, have been investigated. The physical and kinematical parameters in each model are discussed. It is shown that the modeled universe asymptotically becomes isotropic at late times.

We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible K\"ahler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, we show that the Fisher information tensor gives rise to a structure that is sufficiently close to a K\"ahler structure to generalise some classical result on co-adjoint orbits., Comment: Revised and accepted version, 24 pages

We investigate the coupling of the minimal surface equation with a spinor of harmonic type. This arises as the Euler–Lagrange equations of the sum of the volume functional and the Dirac action, defined on an appropriated Dirac bundle. The solutions show a relation to Dirac-harmonic maps under some constraints on the energy-momentum tensor, extending the relation between Riemannian minimal surface and harmonic maps.

The presence of natural fractures in fractured media plays a vital role in the in-situ stress state, which is predominantly influenced by tectonic stresses and local perturbations. Fracture orientation, wellbore stability/orientation, and permeability anisotropy are strongly dependent on local stress variations. In this study, a discrete fracture network (DFN) was generated using the stochastic approach to investigate the correlation between geometrical properties of the fracture network (fracture density and length) and local stress variability. Afterward, the FLAC2D software was employed to propagate the stress redistribution in the simulation process. Considering the tensorial nature of stress, a stress field was calculated under orthogonal static boundary conditions. A tensor-based mathematical formulation was applied to compute total stress variability, shear strain distribution, and total displacement in different fracture network realizations. The results demonstrated that the mean local stress perturbation, effective variance (as an indicator of total stress variability), shear strain distribution, and total displacement increased with the rise of stress ratio and fracture density. The aforementioned parameters decreased as the power-law length exponent of the DFN generator increased. Overall, it is concluded that the stress/strain variability and total displacement in a dense fracture network and large length fractures have their maximum values.