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The first-order resonant Raman spectra of monolayer MoS2 are calculated under the circularly polarized photoexcitation. The anomalously nonzero Raman intensity of the in-plane E mode under the Z ¯ ( σ + σ + ) Z or Z ¯ ( σ − σ − ) Z geometry, which goes against the conventional selection rule, appears under some circumstances when optical absorption occurs at some special reciprocal points between the zone-center Γ and the zone-edge-center M points. At that moment, the valley selectivity to the circular polarization is lifted. The analysis shows that the anomalous Raman intensity of the E mode for the same circularly polarized incident and scattered light is consistent with the pseudo-angular-momentum conservation law. The calculated E Raman tensor of monolayer MoS2 is found to vary with laser energy. The two diagonal terms of the Raman tensor change their signs from mutually opposite to the same when the relative intensity of the in-plane E mode to the out-of-plane A 1 ′ mode increases, indicating the increasingly important role played by the Frolich-type electron-phonon interaction over the deformation potential. Our study may shed new light on the understanding of the novel electron-photon process and assist in the design of new type of optoelectronic devices.

Private individual information are increasingly exposed through high-dimensional and high-order data, with the wide deployment of learning techniques. These data are typically expressed in form of tensors, but there is no principled way to guarantee privacy for tensor-valued queries. Conventional differential privacy is typically applied to scalar values without a precise definition on the shape of the queried data. Realizing that the conventional mechanisms do not take the data structural information into account, we propose Tensor Variate Gaussian (TVG), a new (,δ)-differential privacy mechanism for tensor-valued queries. We further introduce two mechanisms based on TVG with an improved utility by imposing the unimodal differentially-private noise. With the utility space available, the proposed mechanisms can be instantiated with an optimized utility, and the optimization problem has a closed-form solution scalable to large-scale problems. Finally, we experimentally test our mechanisms on a variety of datasets and models, demonstrating that TVG is superior than other state-of-the-art mechanisms on tensor-valued queries.

Tensor factorization methods have recently gained increased popularity. A key feature that renders tensors attractive is the ability to directly model multi-relational data. In this work, we propose ParaSketch, a parallel tensor factorization algorithm that enables massive parallelism, to deal with large tensors. The idea is to compress the large tensor into multiple small tensors, decompose each small tensor in parallel, and combine the results to reconstruct the desired latent factors. Prior art in this di- rection entails potentially very high complexity in the (Gaussian) compression and final combining stages. Adopting sketching matrices for compression, the proposed method enjoys a dramatic reduction in compression complexity, and features a much lighter combining step. Moreover, theoretical analysis shows that the compressed tensors inherit latent identifiability under mild conditions, hence establishing correctness of the overall approach. Numerical experiments corroborate the theory and demonstrate the effectiveness of the proposed algorithm.

A new approximation algorithm is developed to compute the electromagnetic (EM) tensor Green's function in the biaxial anisotropic media based on the Fourier series expansion. First, a rectangular region is chosen as the computation region, in which the EM field can be expressed as a 2-D Fourier series. The Fourier coefficients can be regarded as the EM field in discrete wavenumber domain. We further obtain the EM solution in the spatial domain in the form of Fourier series. Then, we use the finite terms of the Fourier series to approximate the EM field for enhancement of computation efficiency. Because the new method avoids the numerical integration in the infinite wavenumber domain, it is more convenient to implement than other methods based on integral transform. Finally, the spatial distribution of the tensor Green's function for a transversely isotropic medium is presented to verify the proposed approach. The agreements between the results obtained by the present method and the analytic solution demonstrate the validity and the robustness of our algorithm.

Zinc phosphide (Zn3P2) is an II-V compound semiconductor with promising photovoltaic and thermoelectric applications. Its complex structure is susceptible to facile defect formation, which plays a key role in further optimization of the material. Raman spectroscopy can be effectively used for defect characterization. However, the Raman tensor of Zn3P2, which determines the intensity of Raman peaks and anisotropy of inelastic light scattering, is still unknown. In this paper, we use angle-resolved polarization Raman measurements on stoichiometric monocrystalline Zn3P2 thin films and first-principle calculations to obtain Raman tensor of Zn3P2. This has allowed determination of the Raman tensor elements characteristic for the A1g, B1g and B2g vibrational modes. These results have been compared with the theoretically obtained Raman tensor elements and simulated Raman spectra from the lattice-dynamics calculations using first-principles force constants. Excellent agreement is found between the experimental and simulated Raman spectra of Zn3P2 for various polarization configurations, providing a platform for future characterization of the defects in this material.

In this research we simulated how time can be reversed with a rotating strong gravity. At first, we assumed that the time and the space can be distorted with the presence of a strong gravity, and then we calculated the angular momentum density of the rotating gravitational field. For this simulation we used Einstein’s field equation with spherical polar coordinates and the Euler’s transformation matrix to simulate the rotation. We also assumed that the stress-energy tensor that is placed at the end of the strong gravitational field reflects the intensities of the angular momentum, which is the normal (perpendicular) vector to the rotating axis. The result of the simulation shows that the angular momentum of the rotating strong gravity changes its directions from plus (the future) to minus (the past) and from minus (the past) to plus (the future), depending on the frequency of the rotation.

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).

In this paper using the modified couple stress theory, to study the bending, buckling and vibration characteristics of the rectangular Mindlin's nanoplates with graphene material was investigated. With the aim of considering the effects of small scales, the modified couple stress theory, which has only one parameter of length scale and also was presented by Yang in 2002, was used. In the modified couple stress theory; the strain energy density is a function of the components of the strain tensor, curvature tensor, stress tensor and the symmetric part of the couple stress tensor. After obtaining the strain energy, external work, and buckling equation and placing them in the Hamilton's equation, the basic and auxiliary equations of the nanoplates were obtained. Then, by applying boundary and force conditions in the governing equations, the bending, buckling and vibration of the rectangular graphene nanoplates with thickness h and simply-supported conditions were explored. Also, the solution method was the Navier's solution.

Recommendation systems, social network analysis, medical imaging, and data mining often involve processing sparse high-dimensional data. Such high-dimensional data are naturally represented as tensors, and they cannot be efficiently processed by conventional matrix or vector computations. Sparse Tucker decomposition is an important algorithm for compressing and analyzing these sparse high-dimensional data sets. When energy efficiency and data privacy are major concerns, hardware accelerators on resource-constraint platforms become crucial for the deployment of tensor algorithms. In this work, we propose a hybrid computing framework containing CPU and FPGA to accelerate sparse Tucker factorization. This algorithm has three main modules: tensor-times-matrix (TTM), Kronecker products, and QR decomposition with column pivoting (QRP). In addition, we accelerate the former two modules on a Xilinx FPGA and the latter one on a CPU. Our hybrid platform achieves $23.6 \times \sim 1091\times$ speedup and over $93.519\% \sim 99.514 \%$ energy savings compared with CPU on the synthetic and real-world datasets., Comment: 10 pages, 7 figures, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

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.

The problem about steady-state temperature distribution in a homogeneous isotropic medium containing a pore or an insulating inhomogeneity formed by two coalesced spheres of the same radius, under arbitrarily oriented uniform heat flux, is solved analytically. The limiting case of two touching spheres is analyzed separately. The solution is obtained in the form of converged integrals that can be calculated using Gauss-Laguerre quadrature rule. The temperature on the inhomogeneity's surface is used to determine components of the resistivity contribution tensor for the insulating inhomogeneity of the mentioned shape. An interesting observation is that the extreme values of these components are achieved when the spheres are already slightly coalesced.

For many inverse parameter problems for partial differential equations in which the domain contains only well-separated objects, an asymptotic solution to the forward problem involving `polarization tensors' exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the non-linearity. As such, these asymptotic expansions can allow fast and stable reconstruction of small isolated objects. In this paper, we show how such an asymptotic series can be applied to non-linear least-squares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data mist term. Often, the Hessian matrix can play a vital role in dealing with the non-linearity, generating good update directions which accelerate the solution towards a global minimum which may lie in a long curved valley, but computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for non-linearity in the form of lack of superposition in the inverse problem. It does however account for the non-linear saturation of the change in the data with increasing material contrast. We therefore propose to use it as an initial Hessian for quasi-Newton schemes.This is demonstrated for the case of electrical impedance tomography in numerical experimentation, but could be applied to any other problem which has an equivalent asymptotic expansion. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian, providing a proof of principle of the reconstruction scheme.

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.

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus $g$ in the base. These non-flat fibers contribute Kaehler moduli to the four-fold but also add to the three-form cohomology for $g>0$. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional $\mathcal{N}=(1,0)$ superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First, we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively, the non-flat fibers can be avoided by performing birational base changes, analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant., 28 pages, 5 pages appendix, 6 figures, 2 tables

The nonlinear Hall effect is an unconventional response, in which a voltage can be driven by two perpendicular currents in the Hall-bar measurement. Unprecedented in the family of the Hall effects, it can survive time-reversal symmetry but is sensitive to the breaking of discrete and crystal symmetries. It is a quantum transport phenomenon that has deep connection with the Berry curvature. However, a full quantum description is still absent. Here we construct a quantum theory of the nonlinear Hall effect by using the diagrammatic technique. Quite different from nonlinear optics, nearly all the diagrams account for the disorder effects, which play decisive role in the electronic transport. After including the disorder contributions in terms of the Feynman diagrams, the total nonlinear Hall conductivity is enhanced but its sign remains unchanged for the 2D tilted Dirac model, compared to the one with only the Berry curvature contribution. We discuss the symmetry of the nonlinear conductivity tensor and predict a pure disorder-induced nonlinear Hall effect for point groups $C_{3}$, $C_{3h}$, $C_{3v}$, $D_{3h}$, $D_{3}$ in 2D, and $T$, $T_{d}$, $C_{3h}$, $D_{3h}$ in 3D. This work will be helpful for explorations of the topological physics beyond the linear regime., Comment: 9 pages, 3 figures, 1 table. The highlighted tensor elements in Table I can be found here

In this paper, we study the modified gravity theory, where and represent the scalar-torsion and trace of energy-momentum tensor, respectively. According to our cosmological study, we show that the gravity theory has a good agreement with observations. Function type for can be determined by several factors. The most important motivation for determining is that the Lagrangian of model follows the Noether continues symmetry; therefore, according to the Noether approach, we try to determine the function of . Moreover, the proposed model should behave in agreement with observations in terms of phenomenology. As a result, we compared the predictions of the model with observation constrains, for example, the cosmic microwave background and Hubble diagram can be mentioned. In this paper, we study the gravity in the FRW space-time. So, we achieved the effective Lagrangian with respect to independent variables, scale factor , scalar-torsion and trace of energy-momentum tensor . After imposing the Noether symmetry to Lagrangian, we can introduce the suitable form for the function. . Then, we calculated the Noether conservative quantity for the model.

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.

Made available in DSpace on 2021-06-25T11:00:24Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-08-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) We present a five dimensional cosmological metric that reveals a four-dimensional energy-momentum tensor. We analyse three cases for the resulting field equations: null, positive and negative cosmological constant Λ. For the case with null cosmological constant, we obtain a solution able to describe the radiation-dominated era of the universe. The positive five-dimensional cosmological constant case yields a bounce cosmological model. In the negative Λ case, the scale factor for the line element is obtained as a(t)=c5sinh([Formula presented]|Λ|t), where c5 is a constant. This solution can remarkably describe not only the late-time cosmic acceleration but also the non-accelerated stages of the cosmic expansion, namely the radiation and matter dominated epochs in a continuous form. We obtain an analytical equation of state capable of describing the different epochs of the universe and it reads as p=ω(z)ρ with ω(z)=[Formula presented][1−[Formula presented]], with p being the pressure of the universe, ρ its density and z is the redshift. In our model the constant c5 is related to the Hubble constant as H0=[Formula presented]coth[arcsinh([Formula presented])2] and this satisfactorily fits the observational data for the low redshift sample of the experimental measurements of the Hubble parameter, which results in H0=72.2−5.5+5.3 km s−1 Mpc−1 and c5=0.600−0.058+0.061. ITA - Instituto Tecnológico de Aeronáutica - Departamento de Física Universidade de São Paulo Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Rua do Matão 1226, Cidade Universitária Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva - R. Geraldo Alckmin 519 Departamento de Física Instituto de Ciências Ambientais Químicas e Farmacêuticas (ICAQF), Universidade Federal de São Paulo - UNIFESP, Rua São Nicolau no.210, Centro Universidade Estadual Paulista (Unesp), Campus Experimental de Itapeva - R. Geraldo Alckmin 519 FAPESP: 2013/26258-2 FAPESP: 2015/08476-0 FAPESP: 2017/05859-0 FAPESP: 2018/20689-7 CNPq: 313236/2018-6 CNPq: 438562/2018-6 CAPES: 88881.309870/2018-01 CAPES: 88882.446980/2019-01

In this paper, we propose a feasible and effective iteration method for solving the tensor equation A∗NX∗MB=C with a tensor inequality constraint D∗NX∗ME≥F. The global convergence is established. N...

This paper presents a novel direct Jacobian total Lagrangian explicit dynamics (DJ-TLED) finite element algorithm for real-time nonlinear mechanics simulation. The nodal force contributions are expressed using only the Jacobian operator, instead of the deformation gradient tensor and finite deformation tensor, for fewer computational operations at run-time. Owing to this proposed Jacobian formulation, novel expressions are developed for strain invariants and constant components, which are also based on the Jacobian operator. Results show that the proposed DJ-TLED consumed between 0.70x and 0.88x CPU solution times compared to state-of-the-art TLED and achieved up to 121.72x and 94.26x speed improvements in tetrahedral and hexahedral meshes, respectively, using GPU acceleration. Compared to TLED, the most notable difference is that the notions of stress and strain are not explicitly visible in the proposed DJ-TLED but embedded implicitly in the formulation of nodal forces. Such a force formulation can be beneficial for fast deformation computation and can be particularly useful if the displacement field is of primary interest, which is demonstrated using a neurosurgical simulation of brain deformations for image-guided neurosurgery. The present work contributes towards a comprehensive DJ-TLED algorithm concerning isotropic and anisotropic hyperelastic constitutive models and GPU implementation. The source code is available at https://github.com/jinaojakezhang/DJTLED., Comment: Published in International Journal for Numerical Methods in Engineering

We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an "almost symplectic" part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

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 small matrix decomposition of the path integral (SMatPI) is employed to devise expressions for the quantum mechanical amplitude of forward-backward paths in the path integral formulation. The amplitude is expressed as a sum of small matrix products, whose size is equal to that of the system's reduced density matrix, allowing the treatment of composite systems consisting of interacting subunits without the large storage requirements of the full amplitude tensor. Representative applications on a four-spin system with the topology of the basic dendrimer block are presented.

As a generative model , probabilistic linear discriminant analysis (PLDA) has achieved good performance in supervised learning tasks. The model incorporates both within-individual and between-individual variation, and remaining unexplained data variation is assumed to follow Gaussian distribution . However, the assumption of Gaussian distribution makes the model sensitive to the presence of noise and outliers in training set. To address this issue, this paper proposes a robust probabilistic linear discriminant analysis model by assuming Laplace prior on the noise term. Instead of solving high-dimensional linear systems, we embed a Kronecker-decomposable component in the new model for tensor data, significantly reducing the size of problems. As the non-conjugacy of Laplace distribution complicates the calculation of the posteriors of latent variables, we express it to a hierarchical architecture using an Inverse Gamma distribution and then adopt variational expectation–maximization (EM) algorithm to learn model parameters. The reconstruction and classification experiments on several public databases show the superiority of the proposed model compared with the state-of-the-art LDA-based algorithms.

We analyze the learning problem of one-hidden-layer nonoverlapping convolutional neural networks with the rectified linear unit (ReLU) activation function from the perspective of model estimation. The training outputs are assumed to be generated by the neural network with the unknown ground-truth parameters plus some additive noise, and the objective is to estimate the model parameters by minimizing a nonconvex squared loss function of the training data. Assuming that the training set contains a finite number of samples generated from the Gaussian distribution, we prove that the accelerated gradient descent (GD) algorithm with a proper initialization converges to the ground-truth parameters (up to the noise level) with a linear rate even though the learning problem is nonconvex. Moreover, the convergence rate is proved to be faster than the vanilla GD. The initialization can be achieved by the existing tensor initialization method. In contrast to the existing works that assume an infinite number of samples, we theoretically establish the sample complexity of the required number of training samples. Although the neural network considered here is not deep, this is the first work to show that accelerated GD algorithms can find the global optimizer of the nonconvex learning problem of neural networks. This is also the first work that characterizes the sample complexity of gradient-based methods in learning convolutional neural networks with the nonsmooth ReLU activation function. This work also provides the tightest bound so far of the estimation error with respect to the output noise.

We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call “scaffolds”, obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of “moves” on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs.

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.

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 give a broad generalisation of the mapping, originally due to Dennis, Kitaev, Landahl and Preskill, from quantum error correcting codes to statistical mechanical models. We show how the mapping can be extended to arbitrary stabiliser or subsystem codes subject to correlated Pauli noise models, including models of fault tolerance. This mapping connects the error correction threshold of the quantum code to a phase transition in the statistical mechanical model. Thus, any existing method for finding phase transitions, such as Monte Carlo simulations, can be applied to approximate the threshold of any such code, without having to perform optimal decoding. By way of example, we numerically study the threshold of the surface code under mildly correlated bit-flip noise, showing that noise with bunching correlations causes the threshold to drop to $p_{\textrm{corr}}=10.04(6)\%$, from its known iid value of $p_{\text{iid}}=10.917(3)\%$. Complementing this, we show that the mapping also allows us to utilise any algorithm which can calculate/approximate partition functions of classical statistical mechanical models to perform optimal/approximately optimal decoding. Specifically, for 2D codes subject to locally correlated noise, we give a linear-time tensor network-based algorithm for approximate optimal decoding which extends the MPS decoder of Bravyi, Suchara and Vargo., Comment: 23 pages, comments welcome

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution ρ0(x) and iteratively applies the recurrence relation ρk+1(x) = U(x) ρk(x)/∥Uρk∥L1, where U(x) = e-V(x) is defined in terms of the potential energy surface (PES) V(x) with global minimum at x = x*. Upon convergence, the probability distribution becomes a delta function δ(x - x*), so the global minimum can be obtained as the position expectation value x* = Tr[x δ(x - x*)]. QTT representations of V(x) and ρ(x) are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate V(x) for all possible values of x. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain with D = 50 adenine-thymine base pairs, and a discrete non-differentiable potential energy surface, V(p) = mod(N,p), that resolves the prime factors of an integer N, with p in the space of prime numbers {2, 3,..., pmax} folded as a d-dimensional 21 × 22 × ··· × 2d tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.

Robust low-rank tensor completion plays an important role in multidimensional data analysis against different degradations, such as Gaussian noise, sparse noise, and missing entries, and has a variety of applications in image processing and computer vision. In this paper, we investigate the problem of low-rank tensor completion with different degradations for third-order tensors, and propose a transformed tensor nuclear norm method combined the tensor l 1 norm with total variational (TV) regularization. Our model is based on a recently proposed algebraic framework in which the transformed tensor nuclear norm is introduced to capture lower transformed multi-rank by using suitable unitary transformations. We adopt the tensor l 1 norm to detect the sparse noise, and the TV regularization to preserve the piecewise smooth structure along the spatial and tubal dimensions. Moreover, a symmetric Gauss–Seidel based alternating direction method of multipliers is developed to solve the resulting model and its global convergence is established under very mild conditions. Extensive numerical examples on both hyperspectral images and video datasets are carried out to demonstrate the superiority of the proposed model compared with several existing state-of-the-art methods.

Recently the rank-structured tensor approach suggested a progress in the numerical treatment of the long-range electrostatic potentials in many-particle systems and the respective interaction energy and forces [39,40,2]. In this paper, we outline the prospects for tensor-based numerical modeling of the collective electrostatic potential on lattices and in many-particle systems of general type. We generalize the approach initially introduced for the rank-structured grid-based calculation of the collective potentials on 3D lattices [39] to the case of many-particle systems with variable charges placed on $L^{\otimes d}$ lattices and discretized on fine $n^{\otimes d}$ Cartesian grids for arbitrary dimension $d$. As result, the interaction potential is represented in a parametric low-rank canonical format in $O(d L n)$ complexity. The energy is then calculated in $O(d L)$ operations. Electrostatics in large biomolecules is modeled by using the novel range-separated (RS) tensor format [2], which maintains the long-range part of the 3D collective potential of the many-body system represented on $n\times n \times n$ grid in a parametric low-rank form in $O(n)$-complexity. We show that the force field can be easily recovered by using the already precomputed electric field in the low-rank RS format. The RS tensor representation of the discretized Dirac delta [45] enables the efficient energy preserving regularization scheme for solving the 3D elliptic PDEs with strongly singular right-hand side arising in bio-sciences. We conclude that the rank-structured tensor-based approximation techniques provide the promising numerical tools for applications to many-body dynamics, protein docking and classification problems and for low-parametric interpolation of scattered data in data science., Comment: 30 pages, 23 figures

In this paper, we give a new lower bound for the eigenvalues of the Dirac operator defined on the Spin Riemannian hypersurface manifold endowed with 2-tensor, in terms of the Energy-Momentum tensor, scalar curvature and extrinsic curvature. Then this estimate is improved in two different ways by considering the conformal invariance of the Dirac operator. The first is given in term of the first eigenvalue of the Yamabe operator. The latter, is given in terms of the the area of a topological 2-sphere., TUBITAK The Scientific and Technological Research Council of Turkey [120F109]; TUBITAK (The Scientific and Technological Research Council of Turkey), This study was supported by TUBITAK The Scientific and Technological Research Council of Turkey (Project Number: 120F109). Also, The authors would like to appreciate TUBITAK (The Scientific and Technological Research Council of Turkey) for supporting this research.

Using the linearized Vlasov–Maxwell model and the polarization tensor components for weakly magnetized ( $$\left| \omega -{\mathbf {k}}\cdot {\mathbf {v}}\right| >\varOmega $$ ) electron plasma the dispersion relations of parallel and perpendicular propagating electromagnetic modes are evaluated under high frequency/long wavelength limit (i.e., $$\omega >{\mathbf{k }}\cdot {\mathbf{v }}$$ ). The propagation characteristics of R–L and ordinary waves are discussed in the presence of isotropic Fermi–Dirac distribution function in a relativistic regime. The Polylog functions that arise due to arbitrary/partial degeneracy are examined in various degeneracy and relativistic limits. The graphical results are also extracted in the presence of quantum non-degenerate, degenerate and fully degenerate regimes due to the variation in relativistic parameter $$(\frac{T}{ m_{0}c^{2}})$$ , Lorentz factor ( $$\gamma $$ ) and degeneracy parameter $$(\frac{\mu }{T})$$ in weakly and strongly relativistic limits. Furthermore, the derived results are found to be in complete agreement with the previous investigations.

The purpose of this paper is to derive the Green's function of an anisotropic elastic medium and validate it with the effective stiffness tensor of Barnett Shale. We have derived the frequency‐dependent Green's function by using the spectral theorem for matrices, thus simplifying the process of computing Green's function and obtaining analytical solutions. Evaluating the inverse of the Green–Christoffel tensor is an essential part of computing the Green's function. Based on the degeneracy of the eigenvalues of the Green–Christoffel tensor, the inverse of the Green–Christoffel tensor is expressed in the form of partial fractions. Consequently, the stiffness tensors from the measurement of core samples of Barnett Shale are used to validate the Green's function. We use the generalized singular approximation method of effective medium theory to model the effective stiffness of the core samples from microstructural properties. The generalized singular approximation method also allows us to compute the theoretical stiffness tensor of the Barnett Shale for porosity variations. The behaviour of the Green's function, which reflects the behaviour of the media, is studied in the static, low‐ and high‐frequency domains and under different physical parameters. It is observed that the variations of crack‐induced porosity produce different trends in Green's function for vertically transverse isotropic and horizontally transverse isotropic media. Thus, the variation of porosity is observed to be influential in differentiating between transverse isotropic media that have inclusions. The Green's function results presented in this paper have direct applications in the construction of synthetic seismograms for unbounded transversely isotropic media.

We show that if a distribution does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature. The Schouten curvature tensor and the nonholonomicity tensor are used to write the Jacobi equation for the distribution. This leads to a study of second-order optimality conditions for horizontal geodesics in sub-Riemannian geometry. We study conjugate points for horizontal geodesics on the Heisenberg group as an example.

Structured Tucker tensor decomposition models complete or incomplete multiway data sets (tensors), where the core tensor and the factor matrices can obey different constraints. The model includes block-term decomposition or canonical polyadic decomposition as special cases. We propose a very flexible optimization method for the structured Tucker decomposition problem, based on the second-order Levenberg-Marquardt optimization, using an approximation of the Hessian matrix by the Krylov subspace method. An algorithm with limited sensitivity of the decomposition is included. The proposed algorithm is shown to perform well in comparison to existing tensor decomposition methods.

Precision studies of the Higgs boson at future $e^+e^-$ colliders can help to shed light on fundamental questions related to electroweak symmetry breaking, baryogenesis, the hierarchy problem, and dark matter. The main production process, $e^+e^- \to HZ$, will need to be controlled with sub-percent precision, which requires the inclusion of next-to-next-to-leading order (NNLO) electroweak corrections. The most challenging class of diagrams are planar and non-planar double-box topologies with multiple massive propagators in the loops. This article proposes a technique for computing these diagrams numerically, by transforming one of the sub-loops through the use of Feynman parameters and a dispersion relation, while standard one-loop formulae can be used for the other sub-loop. This approach can be extended to deal with tensor integrals. The resulting numerical integrals can be evaluated in minutes on a single CPU core, to achieve about 0.1% relative precision., 16 pages, 4 figures and 1 table

The notation of Moore-Penrose inverse of matrices has been extended from matrix space to even-order tensor space with Einstein product. In this paper, we give the numerical study on the Moore-Penrose inverse of tensors via the Einstein product. More precisely, we transform the calculation of Moore-Penrose inverse of tensors via the Einstein product into solving a class of tensor equations via the Einstein product. Then, by means of the conjugate gradient method, we obtain the approximate Moore-Penrose inverse of tensors via the Einstein product. Finally, we report some numerical examples to show the efficiency of the proposed methods and testify the conclusion suggested in this paper.

In this paper, we study T-eigenvalues of third-order tensors. Definitions of the T-eigenvalues and Hermitian tensors are proposed. We present a commutative tensor family. We prove some T-eigenvalue inequalities for Hermitian tensors, including extensions of Weyl's theorem and Cauchy's interlacing theorem from the matrix case to the tensor case. Finally, we introduce the stability of the T-eigenvalues and study the Lyapunov equation for tensors.

In this paper, pure unilateral support equilibrium (USE) is located among pure Nash and pure Berge equilibrium using tensors. The differences between these equilibria are shown using tensor form of a game and are illustrated with numerical examples. Tensors will help specify the location of each equilibrium using a system of coordinates that brings a solid mathematical foundation of all equilibria and provides the possibility to solve high dimensional problems. A numerical example with a 15-player game is studied to demonstrate the efficiency. Besides, we extend the notion of pure USE to mixed USE when the sets of strategies of all players are finite. We prove a lemma dedicated to inaugurate a method of computing mixed USE profiles. We write corresponding formulas using tensors and their operations, and then we illustrate the new method and lemma method by a numerical example of a 7-player game.

The problem of the joint flow of a turbulent gas stream and a vertically falling wavy liquid film is considered. Tangential and normal stresses on the interfaces are calculated. The components of the Reynolds stress tensor are determined within the framework of the Boussinesq hypothesis. For the case of small Reynolds numbers of a liquid, the problem is reduced to a nonlinear integro-differential equation for the deviation of the layer thickness from the unperturbed level. A numerical study of the evolution of periodic perturbations is carried out. Several typical scenarios of their development are presented.

This paper presents two novel data assimilation (DA) techniques for reconstructing steady turbulent flows at high Reynolds numbers by introducing perturbations to the Reynolds stress tensor compute...

In this paper, we define a (1, 3)-tensor field T(X, Y)Z on Kenmotsu manifolds and give a necessary and sufficient condition for T to be a curvature-like tensor. Next, we present some properties related to the curvature-like tensor T and prove that $$M^{2m+1}$$ is an $$\eta $$ -Einstein–Kenmotsu manifold if and only if $$\sum ^{m}_{j=1}T( \varphi (e_j), e_j) X = 0$$ . Besides, we define a (1, 4)-tensor field t on the Kenmotsu manifold M which determines when M is a Chaki T-pseudo-symmetric manifold. Then, we obtain a formula for the covariant derivative of the curvature tensor of Kenmotsu manifold M. We also find some conditions under which an $$\eta $$ -Einstein–Kenmotsu manifold is a Chaki T-pseudo-symmetric. Finally, we give an example to verify our results and prove that every three-dimensional Kenmotsu manifold is a generalized pseudo-symmetric manifold.

By leveraging neoteric analytical techniques associated with big data, numerous new data-focused computation and service models have flourished in service computing systems. Accurate future predictions based on tensor-based multivariate Markov models can vigorously support enterprise decisions. However, the computation efficiency and quick response of tensor-based multimodal prediction approach are seriously restricted by the curse of dimensionality arising from high-order tensor. Therefore, to alleviate the problem, this paper focuses on proposing a tensor-train (TT)-based computation approach with its scalable implementation for high-order dominant eigen decomposition (HODED) in multivariate Markov models. First, we present a TT-based Einstein product directly based on decomposed TT cores and guarantee that the result remains TT format. Then, we put forward a scalable implementation for TT-based Einstein product in a distributed or parallel manner. Afterwards, we propose a scalable TT-based HODED (TT-HODED) algorithm and a multimodal accurate prediction algorithm. Furthermore, a TT-based big data processing and services framework is presented to provide accurate proactive services. Experimental results based on real-world GPS trajectory dataset demonstrate that TT-HODED algorithm can significantly improve the computation efficiency and reduce the running memory on the premise of guaranteeing the almost consistent prediction accuracy compared to the original HODED algorithm.

Graph-tensor learning operations extend tensor operations by taking the graph structure into account, which have been applied to diverse domains such as image processing and machine learning. However, the running time of graph-tensor operations increases rapidly with the number of nodes and the dimension of data on nodes, making them impractical for real-time applications. In this paper, we propose a GPU library called cuGraph-Tensor for high-performance graph-tensor learning operations, which consists of eight key operations: graph shift (g-shift), graph Fourier transform (g-FT), inverse graph Fourier transform (inverse g-FT), graph filter (g-filter), graph convolution (g-convolution), graph-tensor product (g-product), graph-tensor SVD (g-SVD) and graph-tensor QR (g-QR). cuGraph-Tensor supports scalar, vector, and matrix data processing on each graph node. We propose optimization techniques on computing, memory accesses, and CPU–GPU communications that significantly improve the performance of the graph-tensor learning operations. Using the optimized operations, cuGraph-Tensor builds a graph data completion application for fast and accurate reconstruction of incomplete graph data. In the experiments, the proposed graph learning operations achieve up to 142 . 12 × speedups versus CPU-based GSPBOX and CPU MATLAB implementations running on two Xeon CPUs. The graph data completion application achieves up to 174 . 38 × speedups over the CPU MATLAB implementation, and up to 3 . 82 × speedups with better accuracy over the GPU-based tensor completion in the cuTensor-tubal library.

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.