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A new intrinsically-relativistic kinetic mechanism for generation of non-isotropic relativistic kinetic equilibria in collisionless N-body systems is pointed out. The theory is developed in the framework of the covariant Vlasov statistical description. The new effect is based on the constraints placed by the conservation laws of neutral single-particle dynamics in prescribed background curved-spacetimes demonstrating existence of Killing tensors. As an illustration, the particular case of the Kerr space-time admitting the so-called Carter constant for the particle geodesic motion is considered. The general functional form of the equilibrium kinetic distribution function (KDF) is determined and an explicit realization in terms of Gaussian-like distributions is provided. It is shown that, due to the Carter constant, these equilibrium KDFs exhibit an anisotropic phase-space functional dependence in terms of the single-particle 4-velocity components, giving rise to corresponding non-isotropic continuum fluid fields. The qualitative properties of the equilibrium stress-energy tensor associated with these systems are discussed, with a particular emphasis on the related occurrence of temperature anisotropy effects. The theory is susceptible of astrophysical applications, including in particular the statistical properties of dark matter halos around stellar-mass or galactic-center black holes., 14 pages

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.

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).

In this paper we study the problem of decomposing a given tensor into a tensor train such that the tensors at the vertices are orthogonally decomposable. When the tensor train has length two, and the orthogonally decomposable tensors at the two vertices are symmetric, we recover the decomposition by considering random linear combinations of slices. Furthermore, if the tensors at the vertices are symmetric and low-rank but not orthogonally decomposable, we show that a whitening procedure can transform the problem into the orthogonal case. When the tensor network has length three or more and the tensors at the vertices are symmetric and orthogonally decomposable, we provide an algorithm for recovering them subject to some rank conditions. Finally, in the case of tensor trains of length two in which the tensors at the vertices are orthogonally decomposable but not necessarily symmetric, we show that the decomposition problem reduces to the novel problem of decomposing a matrix into an orthogonal matrix multiplied by diagonal matrices on either side. We provide and compare two solutions, one based on Sinkhorn's theorem and one on Procrustes' algorithm. We conclude with a multitude of open problems in linear and multilinear algebra that arose in our study.

The polarization tensor is a geometric quantity associated with a domain. It is a signature of the small inclusion’s existence inside a domain and used in the small volume expansion method to reconstruct small inclusions by boundary measurements. In this paper, we consider the question of the polarization tensor vanishing structure of general shape. The only known examples of the polarization tensor vanishing structure are concentric disks and balls. We prove, by the implicit function theorem on Banach spaces, that a small perturbation of a ball can be enclosed by a domain so that the resulting inclusion of the core-shell structure becomes polarization tensor vanishing. The boundary of the enclosing domain is given by a sphere perturbed by spherical harmonics of degree zero and two. This is a continuation of the earlier work (Kang, Li, Sakaguchi) for two dimensions.

Given a d-dimensional tensor T:A ×…×A →F (where F is a field), the i-flattening rank of T is the rank of the matrix whose rows are indexed by A , columns are indexed by B =A ×…×A ×A ×…×A and whose entries are given by the corresponding values of T. The max-flattening rank of T is defined as mfrank(T)=max frank (T). A tensor T:A →F is called semi-diagonal, if T(a,…,a)≠0 for every a∈A, and T(a ,…,a )=0 for every a ,…,a ∈A that are all distinct. In this paper we prove that if T:A →F is semi-diagonal, then mfrank(T)≥[Formula presented], and this bound is the best possible. We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an r-uniform multi-hypergraph H are colored with z colors such that each color class is a matching of size t, then H contains a rainbow matching of size t provided z>(t−1)(rtr). This improves previous results of Alon and Glebov, Sudakov, and Szabó., Linear Algebra and its Applications, 625, ISSN:0024-3795, ISSN:1873-1856

Chemical shift tensors in 13C solid-state NMR provide valuable localized information on the chemical bonding environment in organic matter, and deviations from isotropic static-limit powder line shapes sensitively encode dynamic-averaging or orientation effects. Studies in 13C natural abundance require magic-angle spinning (MAS), where the analysis must thus focus on spinning sidebands. We propose an alternative fitting procedure for spinning sidebands based upon a polynomial expansion that is more efficient than the common numerical solution of the powder average. The approach plays out its advantages in the determination of CST (chemical-shift tensor) principal values from spinning-sideband intensities and order parameters in non-isotropic samples, which is here illustrated with the example of stretched glassy polycarbonate.

In this study, we are interested to investigate the dissipative gravitational collapse of anisotropic spherically symmetric radiating star, which satisfies the initially static Karmarkar condition in f ( R , T ) gravity. Moreover, we have chosen a linear model f ( R , T ) = f ( R ) + f ( T ) , where R is Ricci scalar, T is trace of energy-momentum tensor and λ is dimensionless coupling parameter. The terms f ( R ) = R and f ( T ) = 2 λ T describe the geometry and matter of the radiating star, respectively. The complete set of the field equations for the radiating star has been given by the Lagrangian matter L m = − 1 3 ( P R + 2 P ⊥ ) . Firstly, we have established the static core because the collapse proceeds from initially static inner spacetime and further developed by the energy dissipation (non-adiabatically) in terms of heat flux in terms of Vaidya exterior metric. For the anisotropic radiating model, we have verified that the energy conditions and other physical properties are well-behaved in the background of f ( R , T ) theory. Eventually, by setting parameter λ = 0 , the results of our model are restored in General Relativity.

We have achieved the isovector and isoscalar tensor form factors of the N ( 1535 ) state in the framework of the light-cone QCD sum rule approach. In numerical computations, we have considered the most general forms of the interpolating current for the nucleon and the tensor current together with the distribution amplitudes of the N ( 1535 ) resonance. We have shown that the squared momentum transfer, Q 2 , dependence of N ( 1535 ) tensor form factors is characterized by a multipole fit function. Our results provide a perspective of the momentum dependence of the tensor form factors and can help to improve our understanding of the internal structure of nucleon states.

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

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.

Motivated by various physical, cellular and ecological applications, there has been a recent resurgence of interest in studying the boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface, by using “bulk-surface models” (involving volumetric densities and surface densities) or by using models involving dynamical boundary conditions for volumetric densities. The surface in the models has no thickness. However, in some real world applications, the “surface” is actually a thin membrane, having positive, albeit small thickness δ, as in the cases of thermal barrier coatings for turbine engine blades and cell membranes. In such situations, rigorous derivations for these models seem lacking and desirable. In this paper, we start with two full models each of which contains reaction-diffusion equations in the bulk and the thin membrane, respectively, with two types of reasonable transmission conditions linking the two. Then in the limit of δ → 0 , we obtain two effective models, with one being a bulk-surface model and the other being the dynamical boundary value problem model, from which we can also recover the surface density of the other substance. Our analysis reveals that to have such effective models, it is crucial that (i) appropriate transmission conditions are satisfied on the interface between the bulk and the thin membrane, and (ii) the diffusion tensor in the thin membrane is “optimally aligned”, which means the principal axes of the tensor are either normal or tangential to the interface.

The thick cloud coverage phenomenon severely disturbs optical satellite observation missions (covering approximately 40–60% areas in the global scale). Therefore, the manner by which to eliminate thick cloud in remote sensing imagery is greatly significant and indispensable. In this study, we combine the deep spatio-temporal prior with low-rank tensor singular value decomposition (DP-LRTSVD) for thick cloud removal in multitemporal images. On the one hand, DP-LRTSVD utilizes the low-rank characteristic of multitemporal images via the third-order tensor SVD and completion. On the other hand, DP-LRTSVD employs the deep spatio-temporal feature expression ability by 3D convolutional neural network. The proposed framework can effectively eliminate thick cloud in multitemporal images through combining the model-driven and data-driven strategies. Moreover, DP-LRTSVD outperforms on thick cloud removal in the simulated and real multitemporal Sentinel-2/GF-1 experiments compared with model-driven or data-driven methods. In contrast with most methods that can only use a single reference image for thick cloud removal, the proposed method can simultaneously eliminate thick cloud in time-series images.

We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $\mathbb{H}$ in the $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. We describe a decomposition of $T$ into $3$ simple tensors in the $2\times 2\times 2$ case. We also show that the upper bound is the best possible for some of the cases, and we provide various partial results involving tensor decompositions over $\mathbb{C}$ and $\mathbb{H}$., Comment: 24 pages, no figures

In this paper, we discuss the following chemotaxis-Stokes system with nonlinear diffusion and rotation { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 with m > 0 , where Ω is a bounded domain in R 3 , S ( x , n , c ) is a chemotactic sensitivity tensor satisfying S ∈ C 2 ( Ω ¯ × [ 0 , ∞ ) 2 ; R 3 × 3 ) and | S ( x , n , c ) | ≤ ( 1 + n ) − α S 0 ( c ) for all ( x , n , c ) ∈ Ω × [ 0 , ∞ ) 2 with α ≥ 0 and some nondecreasing function S 0 : [ 0 , ∞ ) → [ 0 , ∞ ) . We can show that if m + α > 10 9 , then for any reasonably smooth initial data, the corresponding initial-boundary problem possesses a globally bounded weak solution. The result extends the previous global boundedness result for m + α > 10 9 , α > 0 and m + 5 4 α > 9 8 [15] , and m > 9 8 and α = 0 [24] . Without using the conventional free-energy inequality see [15] , [24] , we use a new iterative bootstrap procedure to estimate the bounds on ∫ Ω n e p for p > 3 2 , which is a crucial step to obtain our main result.

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.

We define new norms for symmetric tensors over ordered normed spaces; these norms are defined by considering linear combinations of tensor products or powers of positive elements only. Relations between the different norms are studied. The results are applied to the problem of representing a finitely exchangeable distribution as a mixture of powers, i.e, mixture of distributions of i.i.d. sequences, using a signed mixing measure., 48 pages

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.

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.

In recent few years, the Gauss-Bonnet f ( G , T ) theory of gravity has fascinated considerable researchers owing to its coupling of trace of the stress-energy tensor T with the Gauss-Bonnet term G . In this context, we focuss ourselves to study bouncing universe with in f ( G , T ) gravity background. Some important preliminaries are presented along with the discussion of cosmological parameters to develop a minimal background about f ( G , T ) theory of gravity. The exact bouncing solutions with physical analysis are provided with the choice of two equation of state parameters. It is shown that the results do agree with the present values of deceleration, jerk and snap parameters. Moreover, it is concluded that the model parameters are quite important for the validity of conservation equation (as the matter coupled theories do not obey the usual conservation law).

A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

We present a six-dimensional $\mathcal{N}=(1,0)$ supersymmetric higher gauge theory in which self-duality is consistently implemented by physically trivial additional fields. The action contains both $\mathcal{N}=(1,0)$ tensor and vector multiplets and is non-trivially interacting. The tensor multiplet part is loosely related to a recently proposed action by Sen that leads to on-shell self-duality in an elegant way. As we also show, Sen's action finds a very natural and direct interpretation from a homotopy algebraic perspective., 29 pages, typos fixed, clarifications added, published version

By a molecular dynamics (MD) simulation method which ensures the system will be under hydrostatic pressure, dynamic and elastic properties of glassy metatstable states are investigated. In the MD method, the simulation cell fluctuates not only in volume but also in shape under constant hydrostatic pressure and temperature. As observed in experiments for many glass forming materials, metastable states in our simulation show a sharp increase in mean-square-displacement at certain temperatures TD. Dynamic heterogeneity is also observed at TD. Elastic properties are calculated from stress and strain relations obtained from the spontaneous fluctuation of internal stress tensor and simulation cell parameters. Each investigated state shows distinctive dynamics while maintaining solid-like elastic properties. The elastic properties stay intact even above TD. It has been shown that the rigidity and mobility of glassy metastable states are compatible under dynamic heterogeneity.

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.

The gauge theory of the de Sitter group, SO$(1,4)$, in the ambient space formalism has been considered in this article. This method is essential to constructing the de Sitter super-conformal gravity and Quantum gravity. $10$ gauge vector fields are needed, corresponding to $10$ generators of the de Sitter group. Using the gauge-invariant Lagrangian, the field equation of these vector fields has been obtained. The gauge vector field solutions are recalled. Then, the spin-$2$ gauge potentials are constructed from the gauge vector field. There are two possibilities for presenting this tensor field: rank-$2$ symmetric and mixed symmetry rank-$3$ tensor fields. To preserve the conformal transformation, a spin-$2$ field must be represented by a mixed symmetry rank-$3$ tensor field, $\mathcal {K}_{\alpha\beta\gamma}$. This tensor field has been rewritten using a generalized polarization tensor field and a de Sitter plane wave. This generalized polarization tensor field has been calculated as a combination of vector polarization, $\mathcal {E}_{\alpha}$, and tensor polarization of rank-$2$, $\mathcal {E}_{\alpha\beta}$, which can be used in the gravitational wave consideration. For the construction of this polarization tensor, the arbitrary constant vector fields appear. We fix it so that, in the limit, $H=0$, one obtains the polarization tensor in Minkowski space-time. It has been shown that under some simple conditions, the spin-$2$ mixed symmetry rank-$3$ tensor field can be simultaneously transformed by the unitary irreducible representation of de Sitter and conformal groups, SO$(2,4)$., Comment: 11 pages, typos corrected

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

Multi-way arrays or tensors are becoming important tools to deal with multidimensional data. This paper is concerned with a one-class classification of second order tensor data. One-class classific...

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.

Fluid kinematics describes the fluid motion without consideration of any force. Classical fluid kinematics adopts Helmholtz velocity decomposition, which is equivalent to Cauchy-Stokes (CS) velocity gradient tensor decomposition. CS decomposes the velocity gradient tensor into a strain-rate (symmetric) tensor and a vorticity (anti-symmetric) tensor. However, several questions arise: (1) since vorticity cannot represent fluid rotation, the vorticity tensor is a mixture of vorticity shear and rigid rotation, (2) since the strain-rate tensor cannot represent fluid shear, the strain-rate tensor is a mixture of stretching and shear, (3) the stretching and shear in the CS decomposition are dependent on the selection of coordinate system and are therefore not Galilean invariant. On the other hand, Liutex is a new physical quantity to represent the rigid fluid rotation and a principal coordinate system can be set up based on Liutex. A principal decomposition of the velocity gradient tensor, or the rotation-stretching-shear decomposition, can be easily carried out in the principal coordinate system with a clear physical meaning, which represents the rigid rotation, stretching (compression) and shear (symmetric and anti-symmetric shear). In the principal decomposition, all elements in three sub-tensors are Galilean invariant and, therefore, the principal decomposition is unique, Galilean invariant and independent of coordinate system. The principal decomposition is then transformed back to the original xyz coordinate system. The Liutex-based principal decomposition creates the new fluid kinematics which is ready for building up new fluid dynamics. Since fluid kinematics is the foundation of the fluid dynamics, the new fluid kinematics could replace the classical fluid kinematics, Helmholtz or CS decomposition, and open a new gate to develop new fluid dynamics especially for vortex science and turbulence research.

A large variety of the samples of novel magnetic materials, which are of high interest due to their exotic properties, are only available in very small sizes. In some cases, it is not possible to synthesize large single crystals; in other cases, the small size itself is the key prerequisite to manifest a specifically interesting property of the material. The smallness of a sample rises a problem of the detection of the static magnetic response and of the electron spin resonance (ESR) signal. To overcome this problem, we propose to use a cantilever-based (torque-detected) setup with the capability of a simultaneous measurement of ESR and static magnetization. This setup offers a high sensitivity and the ability to acquire along with the ESR signal the components of the magnetization tensor in a single experimental run. Here, we present the working principle of this setup, as well as the estimate of its sensitivity from the measurements on the standard Co Tutton salt sample.

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.

It has been confirmed by comparison with particle velocities of surface waves of wave normal perpendicular to borehole axis measured by collocated seismic broadband array or borehole seismometers that high-sensitivity borehole tensor strainmeters could measure surface strain waves qualitatively. However, quantitative measurability of strain waves of wave normal being oblique to borehole axis by borehole tensor strainmeters is still a challenging problem. In this article, we introduce a scattering model of oblique-incidence seismic waves by an empty borehole to investigate the influence of incident-wave normal and polarization on theoretical-frequency-response bandwidth of borehole tensor strainmeters. Within the theoretical bandwidth of borehole tensor strainmeters, zero-frequency gains are introduced to build quantitative relation between instrumental strain waves of borehole tensor strainmeters and oblique-incidence seismic strain waves. Better understanding the influence of incident-wave normal and polarization on the relation between instrumental strain and strain components induced by the incident strain wave shows that incident-wave normal and polarization must be assessed by other instrumentation to promise quantitative measurement of incident strain waves by borehole tensor strainmeters, which would benefit in situ calibration of borehole tensor strainmeters.

In this paper, we study a chemotaxis-consumption system { u t = ∇ ⋅ ( ∇ u − u S ( x , u , v ) ∇ v ) + f ( u ) , v t = Δ v − u v , under no-flux boundary conditions in a smooth bounded domain Ω ⊂ R n where the chemotactic sensitivity tensor S ∈ C 2 ( Ω ‾ × [ 0 , + ∞ ) 2 ; R n × n ) fulfills that there exists some nondecreasing function S 0 on [ 0 , + ∞ ) such that | S ( x , u , v ) | ⩽ S 0 ( v ) for all ( x , u , v ) ∈ Ω ‾ × [ 0 , + ∞ ) × [ 0 , + ∞ ) and f ∈ C 1 ( [ 0 , + ∞ ) ) satisfies f ( 0 ) = 0 as well as lim s → + ∞ ⁡ f ( s ) s = − ∞ , which generalize the standard logistic source f ( s ) = ρ s − μ s l , ρ ∈ R , μ > 0 , l > 1 . It is show that for all sufficiently smooth initial data the system possesses a global-in-time solution in an appropriately generalized sense which indicates generalized logistic source can rule out blow-up for such kind of models. Our finding significantly improves existing global solutions in related chemotaxis-consumption systems.

The J-integral and stress intensity factor (SIF) are two major parameters in linear elastic fracture mechanics (LEFM) for the fracture criterion. In this paper, we focus on the J-integral of the slant crack and the slant rigid-line inclusion under the remote anti-plane shear. By employing the degenerate kernel, the path independence of J-integral is analytically demonstrated by using the elliptic coordinates. The positive and negative J-integrals are also analytically derived and numerically implemented by using the dual BEM for the crack and the rigid-line inclusion, respectively. It is interesting to find that the J-integral is not an invariant by using different observer systems but is one component of the vector of the first order tensor. Transformation law of the J-integral with respect to different observers is analytically proved and numerically demonstrated. Finally, the tensor property of order one is examined.

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.

Closed-form expressions for poroelastic coefficients are derived for anisotropic materials exhibiting single and double porosity. A novel feature of the formulation is the use of the principle of superposition to derive the governing mass conservation equations from which analytical expressions for the Biot tensor and Biot moduli, among others, are derived. For single-porosity media, the mass conservation equation derived from the principle of superposition is shown to be identical to the one derived from continuum principle of thermodynamics, thus confirming the veracity of both formulations and suggesting that this conservation equation can be derived in more than one way. To provide further insight into the theory, numerical values of the poroelastic coefficients are calculated for granite and sandstone that are consistent with the material parameters reported by prominent authors. In this way, modelers are guided on how to determine these coefficients in the event that they use the theory for full-scale modeling and simulations.

Analysis of the failure mechanism of inclined coal pillars is one of the complicated issues. The wide variability of dip angles of inclined coal pillars makes it more complex. The asymmetric stress distribution and the tendency of shearing along the bedding planes make the inclined coal pillars to behave differently from the flat coal pillars. There is a need for in-depth investigation of the failure mechanism for addressing the instability problems of the inclined coal pillars. Most of the literature quantifies only the magnitudes of the mean principal stresses by classical statistics. As the stress is a second-order tensor having six independent components, the classical statistics is not appropriate to calculate the mean and variability of the principal stresses at the onset of failure of the pillars. In this paper, a comprehensive analysis is done to understand the complex failure mechanism of the inclined coal pillar using numerical modelling as well as tensorial statistics and validated the results with field measurement data of failure cases. The failure mechanism is analysed by quantification of the characteristics of the inclined coal pillars by the principal-stress magnitude and its orientation, induced at the time of failure. Since the spatial variability of the magnitudes and orientations of the induced principal stresses exist within the inclined coal pillars, the mean induced principal stresses $$\left( {\overline{{\sigma_{1} }} {, }\;\overline{{\sigma_{2} }} {\text{ and }}\overline{{\sigma_{3} }} } \right)$$ are used to quantify the stress states within it. The failure stress states within the coal pillars having different dip angles are generated by the calibrated elasto-plastic numerical modelling with the ubiquitous joint model. Several statistical parameters are calculated to quantify the stress-tensor variability and the correlation among the stress-tensor components. It is found that the correlation coefficients among the shear components increase significantly with the increase of the coal pillar dip angle. Therefore, the inclined coal pillars are highly susceptible to shear failure. The magnitudes, as well as orientations of the mean induced principal stresses within the coal pillars obtained through numerical modelling, are quantified by the tensorial as well as classical statistics. It is found that the magnitude of the mean major induced principal stress $$\left( {\overline{{\sigma_{1} }} } \right)$$ at the time of failure, i.e. the strength of the pillar decreases with the increase of the dip angles. The validation of the results with the actual stress measurement data shows that all the failed pillar cases are correctly predicted by the tensorial statistical approach whereas the classical statistical approach does not effectively predict the actual failed condition of the pillars. The study would help to characterise the behaviour of the inclined pillars and address the instability issues for safe and efficient mining of inclined coal seams.

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.

This work presents the first simulations of the full optical rotation (OR) tensor at coupled cluster with single and double excitations (CCSD) level in the modified velocity gauge (MVG) formalism. The CCSD-MVG OR tensor is origin independent, and each tensor element can in principle be related directly to experimental measurements on oriented systems. We compare the CCSD results with those from two density functionals, B3LYP and CAM-B3LYP, on a test set of 22 chiral molecules. The results show that the functionals consistently overestimate the CCSD results for the individual tensor components and for the trace (which is related to the isotropic OR), by 10%-20% with CAM-B3LYP and 20%-30% with B3LYP. The data show that the contribution of the electric dipole-magnetic dipole polarizability tensor to the OR tensor is on average twice as large as that of the electric dipole-electric quadrupole polarizability tensor. The difficult case of (1S,4S)-(-)-norbornenone also reveals that the evaluation of the former polarizability tensor is more sensitive than the latter. We attribute the better agreement of CAM-B3LYP with CCSD to the ability of this functional to better reproduce electron delocalization compared with B3LYP, consistent with previous reports on isotropic OR. The CCSD-MVG approach allows the computation of reference data of the full OR tensor, which may be used to test more computationally efficient approximate methods that can be employed to study realistic models of optically active materials.

The contact network, defined by a fabric tensor, influences phenomena such as force or heat transfer in a granular assembly. The correlation between fabric, stress, and conductivity has, however, been least explored. Furthermore, a link between these quantities may help to build a benchmark method to assist experiments substantially. The present work bridges the gap between the macroscopic quantities, such as stress and conductivity tensor, using its microscopic connection and the fabric tensor. The study presents a few interesting functional forms and non-dimensional macroscopic quantities that couple conductivity, stress, and fabric. The main feature of these functional forms is that they are independent of friction. These functional forms will provide a good estimate of fabric and conductivity which can be used real-time during the experimental investigations as well.

In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras., 32 pages, comments are welcome

We present a second-order recursive Fermi-operator expansion scheme using mixed precision floating point operations to perform electronic structure calculations using tensor core units. A performance of over 100 teraFLOPs is achieved for half-precision floating point operations on Nvidia's A100 tensor core units. The second-order recursive Fermi-operator scheme is formulated in terms of a generalized, differentiable deep neural network structure, which solves the quantum mechanical electronic structure problem. We demonstrate how this network can be accelerated by optimizing the weight and bias values to substantially reduce the number of layers required for convergence. We also show how this machine learning approach can be used to optimize the coefficients of the recursive Fermi-operator expansion to accurately represent fractional occupation numbers of the electronic states at finite temperatures., Comment: 10 pages, 6 figures

Mathematical models are developed for planar symmetric slot antennas (SAs) with constant slot width using a tensor Green function that takes into account both principal and cross-polarization components in the far-field zone. It is shown that such models can be employed in the study of the electrodynamic properties of SAs with different configurations of slots (linear, exponentially increasing, etc.). The mathematical models are used to analyze the effect of geometrical sizes of the slots in the SAs under study on the radiation patterns (RPs). It is shown that the calculated RPs are in good agreement with the experimental results that makes it possible to reveal applicability limits of the proposed mathematical models. The SAs are studied in the microwave range at a central frequency of 10 GHz.

In this article, we study a tensor-based multitask learning (MTL) method for classification. Taking into account the fact that in many real-world applications, the given training samples are limited and can be inherently arranged into multidimensional arrays (tensors), we are motivated by the advantages of MTL, where the shared structural information among related tasks can be leveraged to produce better generalization performance. We propose a regularized tensor-based MTL method for joint feature selection and classification. For feature selection, we employ the Fisher discriminant criterion to both select discriminative features and control the within-class nonstationarity. For classification, we take both shared and task-specific structural information into consideration. We decompose the regression tensor for each task into a linear combination of a shared tensor and a task-specific tensor and propose a composite tensor norm. Specifically, we use the scaled latent trace norm for regularizing the shared tensor and the $\ell _{1}$ -norm for task-specific tensor. Further, we give a computationally efficient optimization algorithm based on the alternating direction method of multipliers (ADMMs) to tackle the joint learning of discriminative features and multitask classification. The experimental results on real electroencephalography (EEG) datasets demonstrate the superiority of our method over the state-of-the-art techniques.