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Let M be a real hypersurface of a complex space form Mn(c)$M^n(c)$, c≠0$c\ne 0$. Suppose that the structure vector field ξ of M is an eigen vector field of the Ricci tensor S, Sξ=βξ$S\xi =\beta \xi$, β being a function. We study on M, a gradient pseudo‐Ricci soliton (M,g,f,λ,μ$M,g,f,\lambda ,\mu$) that is an extended concept of gradient Ricci soliton, closely related to pseudo‐Einstein real hypersurfaces. When n≥3$n\ge 3$, we show that M is a Hopf hypersurface. [ABSTRACT FROM AUTHOR]

In this review article, we revisit the topic of baryogenesis, which is the physical process that generated the observed baryon asymmetry during the first stages of the primordial Universe. A viable theoretical explanation to understand and investigate the mechanisms underlying baryogenesis must always ensure that the Sakharov criteria are fulfilled. These essentially state the following: (i) baryon number violation; (ii) the violation of both C (charge conjugation symmetry) and CP (the composition of parity and C); (iii) and the departure from equilibrium. Throughout the years, various mechanisms have been proposed to address this issue, and here we review two of the most important, namely, electroweak baryogenesis (EWB) and Grand Unification Theories (GUTs) baryogenesis. Furthermore, we briefly explore how a change in the theory of gravity affects the EWB and GUT baryogenesis by considering Scalar–Tensor Theories (STT), where the inclusion of a scalar field mediates the gravitational interaction, in addition to the metric tensor field. We consider specific STT toy models and show that a modification of the underlying gravitational theory implies a change in the time–temperature relation of the evolving cosmological model, thus altering the conditions that govern the interplay between the rates of the interactions generating baryon asymmetry, and the expansion rate of the Universe. Therefore, the equilibrium of the former does not exactly occur as in the general relativistic standard model, and there are consequences for the baryogenesis mechanisms that have been devised. This is representative of the type of modifications of the baryogenesis processes that are to be found when considering extended theories of gravity. [ABSTRACT FROM AUTHOR]

The phenomenon of buckling of a nonlinearly elastic hollow circular cylinder with dislocations under the action of hydrostatic pressure is studied. The tensor field of the density of continuously distributed dislocations is assumed to be axisymmetric. The subcritical state is described by a system of nonlinear ordinary differential equations. To search for equilibrium positions that differ little from the subcritical state, the bifurcation method is used. Within the framework of the model of a compressible semi-linear (harmonic) material, the critical pressure at which the loss of stability occurs is determined, and the buckling modes are investigated. The effect of edge dislocations on the equilibrium bifurcation is analyzed. It is shown that the loss of stability can also occur in the absence of an external load, i.e., due to internal stresses caused by dislocations. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the geometry of n-dimensional differentiable manifolds endowed with F(a0, a1, ...., aan)-structure satisfying anFn +an-1Fn-1+.......+a1V F+a0I = 0 and establish its existence. Also, it is proved that for the complex numbers, the dimension of a manifold M endowed with F(a0, a1, ...., aan)-structure is even. Furthermore, we study the Nijenhuis tensor of a tensor field F of type (1,1) satisfying the general quadratic equation, which is a particular case of the F(a0, a1, ......, aan)-structure. At last, we study the integrability conditions of an F(a0, a1, ......, aan)-structure. [ABSTRACT FROM AUTHOR]

Complex systems are generating more and more high-dimensional data for which tensor analysis showed promising results by capturing complex correlation structures of data. Such data is often distributed among various sites creating challenges for developing data-driven models. Specifically, data privacy and security concerns have been exacerbated in recent years and drove the demand to store and analyze data at the edge of networks rather than sharing it with a centralized server. Federated learning frameworks have been introduced as a solution to these concerns. These frameworks allow local clients to learn local models and collaborate with others to develop a more generalizable aggregated model while handling data privacy issues. In this article, we propose a federated generalized scalar-on-tensor regression framework where multiple local tensor models are learned at the edge, and their parameters are shared with and updated by an aggregator. Experiments on synthetic data sets and two real-world data sets from agriculture and manufacturing domains show the superiority of our approach over several benchmarks. [ABSTRACT FROM AUTHOR]

We show that a three rank Lanczos type tensor field is an appropriate choice to describe relativistic electromagnetic and gravitational effects. More precisely, we identify the irreducible field-decompositions of this tensor as gravitational and electromagnetic fields. A set of divergence equations are proposed as field equations for the unified field. [ABSTRACT FROM AUTHOR]

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y (s 1 , ... , s k) with k ≥ 2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k = 2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic s p (2 k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the s u (1 , 1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y (s 1 , s 2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s 1 , s 2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1 , 1) is obtained. [ABSTRACT FROM AUTHOR]

The almost contact metric structure that we have on a real hypersurface M in the complex quadric Qm = SOm+2/SOmSO2 allows us to define, for any nonnull real number k, the k-th generalized Tanaka-Webster connection on M, ∇ ^ (k) . Associated to this connection, we have Cho and torsion operators F X (k) and T X (k) , respectively, for any vector field X tangent to M. From them and for any symmetric operator B on M, we can consider two tensor fields of type (1,2) on M that we denote by B F (k) and B T (k) , respectively. We classify real hypersurfaces M in Qm for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator Lξ on M. [ABSTRACT FROM AUTHOR]

We consider gradient flows of surface energies that depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank–Oseen–Helfrich energy. [ABSTRACT FROM AUTHOR]

Modeling the velocity gradient dynamics in incompressible turbulence requires modeling two unclosed quantities: the pressure Hessian tensor and the viscous Laplacian tensor. In this work, we model the pressure Hessian tensor using a combination of two different physics-embedded deep neural networks. The first neural network is trained specifically to predict the alignment tendencies of the eigen-vectors of the pressure Hessian tensor, whereas the second neural network is trained only to predict the magnitude of the tensor. This separation of tasks allows us to define mathematically optimal and physics-informed customized loss functions separately for the two aspects (alignment and magnitude) of the tensor. Both neural networks take invariants of the velocity gradient tensor as inputs. Even though the training of the two networks is performed using direct numerical simulation database of an incompressible stationary isotropic turbulence at a particular Reynolds number, we extensively evaluate the model at different Reynolds numbers and in different kinds of flow fields. In incompressible flows, the proposed model shows significant improvements over the existing phenomenological model (the recent fluid deformation closure model or the RFD model) of the pressure Hessian tensor. While the improvements in the alignment tendencies are convincingly evident in the shapes of the probability density functions of the cosines of various angles between eigenvectors, the improvements in the prediction of the magnitude of the pressure Hessian tensor using the new model are quantifiable in the range of 28%–89% (depending on the type of the flow field) compared to the RFD model. [ABSTRACT FROM AUTHOR]

In this work, we utilize the dynamic invariant method to obtain a solution for the time-dependent Schrödinger equation, aiming to explore the quantum theory of a p-form gauge field propagating in D-dimensional de Sitter spacetimes. Thus, we present a generalization, through the use of p-form gauge fields, of the quantization procedure for the scalar, electromagnetic, and Kalb–Ramond fields, all of which have been previously studied in the literature. We present an exact solution for the p-form gauge field when D=2(p+1), and we highlight the connection of the p=4 case with the chiral N=2, D=10 superstring model. We could observe particle production for D≠2(p+1) because the solutions are time-dependent. Additionally, observers in an accelerated co-moving reference frame will also experience a thermal bath. This could have significance in the realm of extra-dimensional physics, and presents the intriguing prospect that precise observations of the Cosmic Microwave Background might confirm the presence of additional dimensions.

This study proposed a matched field source localization method based on tensor decomposition. By considering the advantages of tensors in multidimensional data processing, a three‐dimensional tensor signal model of space‐time‐frequency is constructed, and the signal subspace is estimated using high‐order singular value decomposition. The source position is estimated by matching the measured data tensor signal subspace with the replica field tensor signal subspace. The S5 event data of SWellEx‐96 is processed by the proposed tensor‐based matched‐field processing (TMFP). The comparison with the results of conventional matched field processing shows that TMFP has a better suppression effect on ambient noise under low SNR and better source localization performance. [ABSTRACT FROM AUTHOR]

This paper aims to explore the metallic structure J 2 = p J + q I , where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle T M over almost quadratic ϕ -structures (briefly, (ϕ , ξ , η) ). Tensor fields F ˜ and F * are defined on T M , and it is shown that they are metallic structures over (ϕ , ξ , η) . Next, the fundamental 2-form Ω and its derivative d Ω , with the help of complete lift on T M over (ϕ , ξ , η) , are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures F ˜ and F * are determined using complete and horizontal lifts on T M over (ϕ , ξ , η) , respectively. Finally, we prove the existence of almost quadratic ϕ -structures on T M with non-trivial examples. [ABSTRACT FROM AUTHOR]

We compute conformally covariant actions and operators for tensors with mixed symmetries in arbitrary dimension d. Our results complete the classification of conformal actions that are quadratic on arbitrary tensors with three indices, which allows to write corresponding conformal actions for all tensor species that appear in the decomposition of the distorsion tensor of an arbitrary metric-affine theory of gravity including both torsion and nonmetricity. We also discuss the degrees of freedom that such theories are propagating, as well as interacting metric-affine theories that enjoy the conformal actions in the Gaussian limit. [ABSTRACT FROM AUTHOR]

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex. [ABSTRACT FROM AUTHOR]

There is an unavoidable association of traversability of wormholes to the violation of null energy condition which in turn indicates the presence of exotic or non-exotic matter in the wormhole geometry. The exotic matter possesses the negative energy that is required to sustain the wormhole. Recently studies are done to solve this problem so as to avoid the exotic matter. In this work, we attempt to find such solution in the framework of F (R , T) gravity where F (R , T) = R + 2 λ T , here R and T are Ricci scalar and trace of energy momentum tensor respectively, using the Yukawa–Casimir shape function. For Yukawa–Casimir wormhole, it is assumed that the exotic energy is sourced from the Casimir energy density. We have examined the energy conditions using Yukawa–Casimir shape function b (r) = 2 r 0 3 + r 0 2 3 r exp − μ r − r 0 , where μ is a positive mass scale. We have taken different values of μ to study the role of Yukawa–Casimir energy in formation of traversable wormholes. The equilibrium aspect of the model is also investigated with the help of TOV equation. [ABSTRACT FROM AUTHOR]

An intensity tensor of quadrupole doublets and an electric field gradient tensor for Fe3+ at M1 sites in aegirine–augite ((Ca0.16Na0.86)∑1.02(Mg0.13Fe2+0.04Fe3+0.72 Al0.07)∑0.96Si2.01O6) are determined using single-crystal Mössbauer spectroscopy. The components of the intensity tensor are IXX = 0.670 (19), IYY = 0.353 (14), IXY = −0.113 (37) and IZZ = 0.477 (33). The components of the electric field gradient tensor (VXX, VYY and VZZ) for Fe3+ at M1 sites in aegirine–augite are −5.96 × 109, −4.65 × 1010 and 5.23 × 1010 C/m3, respectively. Comparisons of the intensity tensor of aegirine–augite with those of aegirine and augite (Wo40En45Fs16) that have already been reported and the IXX, IYY, IXY and IZZ intensity tensor components of aegirine–augite in this study are almost the same as those of aegirine, but different from those of augite. While the M2 sites of aegirine–augite and aegirine are fully occupied with Na+ and Ca2+ ions, the M2 sites of augite are not fully occupied with Ca2+. The compositional dependency of the intensity tensor components suggests that the intensity tensor components for Fe3+ at the M1 site of a solid solution between aegirine and augite are dependent on the occupancy of large cations such as Ca2+ and Na+ at M2 sites. [ABSTRACT FROM AUTHOR]

In this paper, we examine the differential-geometric aspect of constant-rank mappings of smooth manifolds based on the concept of a graph as a smooth submanifold in the space of the direct product of the original manifolds. The nonmaximality of the rank provides the fibered nature of the graph. A Riemannian structure on manifolds enriches the geometry of the graph, which now essentially depends on the induced field of the metric tensor; we characterize relatively affine, projective, and g-umbilical mappings. The final part of the paper is devoted to mappings of Euclidean spaces of the types described earlier in terms of V. T. Bazylev's constructive graph. [ABSTRACT FROM AUTHOR]

We derive a general expression for the trace of the energy–momentum tensor -deformed field theories using a dynamical change of coordinates. Then we perform a dimensional reduction of the bilinear operator and obtain a new -like deformation of the quantum mechanics of free nonrelativistic fermions and the interacting Calogero–Sutherland system. The deformation leads to a change in the energy spectrum but does not affect the eigenfunctions. Furthermore, an expression for the deformed classical Lagrangian is obtained. We also study the correspondence between the two-dimensional Yang–Mills theory and the Calogero–Sutherland system in the presence of the deformation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the localization of five-dimensional tensor gauge fields called Kalb–Ramond fields in the modified teleparallel braneworld models. The aforementioned branes are generated by gravity coupled to a real scalar field with nonstandard kinetic terms. For two specific forms of the noncanonical expressions, background scalar field solutions are particularly discussed in the presence of different forms of f (T) gravity. By applying nonminimal coupling functions in the tensor gauge field action, it is shown that massless zero modes of Kalb–Ramond gauge fields can be localized on the f (T) branes with noncanonical scalar fields. We also find that potentials of Kalb–Ramond massive modes are volcano-like regarding the equation of motion in the supersymmetric quantum mechanic scenario. Furthermore, the effects of parameters that control the deviation of the usual teleparallel theory are considered on the massless Kaluza–Klein mode and effective potential. [ABSTRACT FROM AUTHOR]

In this paper, we study the holographic dual to an asymptotically anti-de Sitter black brane in an Einstein–Yang–Mills model with a non-minimal coupling between the Riemann and Yang–Mills fields. First, we construct a planar black hole solution of this model up to the first order of the non-minimal coupling of the Yang–Mills field with the Riemann–Christoffel tensor, denoted as q 2. Then, we calculate the color non-abelian direct current conductivity and the ratio of shear viscosity to entropy density for this solution. Our result for the shear viscosity η to entropy density s ratio saturates the Kovtun, Son, and Starinets bound, which is proportional to 1 4 π . However, our result for the conductivity is new up to the first order of q 2. [ABSTRACT FROM AUTHOR]

We propose a reduced-order method for the elastodynamic homogenization of periodic composites. With the help of Bloch-wave expansion and Green's function, the Lippmann–Schwinger equations relating the dynamic field variable tensors of strain, velocity, stress and linear momentum are obtained. Then the constitutive relation of the averaged dynamic field tensors and the dispersion relation between frequency and wave vector in Willis theory are formulated explicitly. Using the data compression algorithm, k-means clustering, we decompose computational domain into clusters of possibly disjoint cells. The Lippmann–Schwinger equations are then discretized and solved efficiently. Numerical tests for three-dimensional particle-reinforced composites verify the effectiveness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The paper is devoted to the electromagnetic inverse scattering problem for a dielectric anisotropic and magnetically isotropic media. The properties of an anisotropic medium with respect to electromagnetic waves are defined by the tensors, which give the relation between the inductions and the fields. The tensor Fourier diffraction theorem derived in the paper can be considered a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The method is based on the first Born approximation. [ABSTRACT FROM AUTHOR]

We have investigated the fermionic spectral function in p-wave holographic superconductors. We show that the vector model with minimal coupling reveals a p-wave spectral function with Fermi arc. This should be contrasted with the previous investigation where p-wave arc was demonstrated in the presence of a tensor field. We study the momentum dependent order parameter, the ω-gap in the real part of the conductivity and the fermion spectral function. In addition, we juxtapose the fermionic spectral gap with the order parameter in the holographic set. We demonstrate the impact of coupling constants, temperature and chemical potential on the spectral function. [ABSTRACT FROM AUTHOR]

This article introduces the concept of near-field (NF) matching as a continuum-mode generalization of port matching in circuit theory suitable for field-theoretic electromagnetic energy transfer scenarios, with a focus on spatio-frequency processes in coupled systems. The concept is rigorously formulated using the full electromagnetic Green's function of a generic receiving surface interacting with arbitrary illumination fields where the Riemannian structure and the electromagnetic boundary condition of the problem are encoded into the tensor structure of a Green's function on a manifold. After a carefully selected combination of proper function spaces for the various field quantities involved, we utilize exact methods to estimate the sizes of various operator quantities using the appropriate function space norms. A field-theoretic measure of power transfer efficiency in generalized NF matching scenarios is introduced, and exact upper bounds on this efficiency are derived using Young's inequality for integral kernel operators. This theoretical study complements and generalizes the largely empirical and problem-specific literature on wireless energy transfer by providing an exact and rigorous mathematical framework that can guide and inform future optimization and design processes. [ABSTRACT FROM AUTHOR]

A theoretical investigation into the hidden source field of mutation affecting the curvature change in the spike (S) protein of Omicron BA.5 and related variants is reported. The curvature in the open string shape of S protein is defined using the Yang-Mills field over a new type of connection known as quantum genotype. By adding more invariant properties of curvature two-forms to the adaptive tensor fields in DNA, RNA, and protein molecules, we redefined the hidden quantum biological co-state of the mutational field between the virus and the host cell. The new algorithm was applied to classify mutations in S protein with a support vector machine. The results showed that the average performance of the prediction of unknown amino acids in 14 variants including Omicron BA.1-BA.5 is 97.79%. Additionally, we demonstrated a new approach for the quantitative measurement of changing curvature of S protein mutations in each amino acid. The empirical analysis of the probability distribution of time series data showed the evolution of the quantum genotype over time, revealing a new direction of evolution in SARS-CoV-2's quantum genotype opposite to the period between 2020 and 2021. This work can be applied to detect new incoming novel variants of SARS-CoV-2 in the future and provide insight into the coupling between the passive and active sides of communication in the biomolecular layer. [ABSTRACT FROM AUTHOR]

We develop the BRST–BV approach to the construction of the general off-shell Lorentz covariant cubic, quartic, and -tic interaction vertices for irreducible higher-spin fields on -dimensional Minkowski space. We consider two different cases for interacting integer higher-spin fields with both massless and massive fields. The deformation procedure to find a minimal BRST–BV action for interacting higher-spin fields, defined with help of a generalized Hilbert space, is based on the preservation of the master equation in each power of the coupling constant starting from the Lagrangian formulation for a free gauge theory. For illustration, we consider the construction of local cubic vertices for irreducible massless fields of integer helicities, and massless fields and one massive field of spins . For a triple of two massless scalars and a tensor field of integer spin, the BRST–BV action with cubic interaction is explicitly found. In contrast to the previous results on cubic vertices, following our results for the BRST approach to massless fields, we use a single BRST–BV action instead of the classical action with reducible gauge transformations. The procedure is based on the complete BRST operator that includes the trace constraints used in defining the irreducible representation with a definite integer spin. [ABSTRACT FROM AUTHOR]

By replacing the internal energy with the free energy, as coordinates in a "space of observables", we slightly modify (the known three) non-holonomic geometrizations from Udriste's et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R 4 as the "graph" of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general "equivalence problem" between classical entropy and statistical entropy. [ABSTRACT FROM AUTHOR]

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics. [ABSTRACT FROM AUTHOR]

A new model of the regular black hole in (2+1)−$(2+1)-$dimensions is introduced by considering an appropriate matter field as the energy‐momentum tensor. First, we propose a novel model of d‐dimensional energy density that in (2+1)−$(2+1)-$dimensions leads to the existence of an upper bound on the radius of the event horizon and a lower bound on the mass of the black hole which are motivated by the features of astrophysical black holes. According to these bounds, we introduce an admissible domain for the event horizon radius, depending on the metric parameters. After investigation of geometric properties of the obtained solutions, we study the thermal stability of the solution in the canonical ensemble and find that the regular black hole is thermally stable in the mentioned admissible domain. Besides, the free energy is calculated to examine the global stability of the solution. A new model of the regular black hole in (2 + 1)−dimensions is introduced by considering an appropriate matter field as the energy‐momentum tensor. First, we propose a novel model of d‐dimensional energy density that in (2 + 1)−dimensions leads to the existence of an upper bound on the radius of the event horizon and a lower bound on the mass of the black hole which are motivated by the features of astrophysical black holes. According to these bounds, we introduce an admissible domain for the event horizon radius, depending on the metric parameters. After investigation of geometric properties of the obtained solutions, we study the thermal stability of the solution in the canonical ensemble and find that the regular black hole is thermally stable in the mentioned admissible domain. Besides, the free energy is calculated to examine the global stability of the solution. [ABSTRACT FROM AUTHOR]

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ = 1 , 2 , 3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition. [ABSTRACT FROM AUTHOR]

Background: Obtaining accurate displacement measurements for large material deformation and/or rotation presents a distinct challenge to digital image correlation (DIC) due to cumulative and decorrelation errors, particularly near material boundaries. Objective: We aim to accurately measure the deformation gradient tensor near boundary discontinuities in situations of large deformation and large deformation gradients. Methods: To achieve this goal, the locations of randomly distributed particles are tracked using an open-source particle-tracking software, Trackpy. A least-squares estimate of the deformation gradient tensor field uses nearest-neighbor material vectors and a first-order Finite Difference (FD) approximation, circumventing common errors in other methods. The error caused by FD approximation and that incurred by measurement are derived and tested with exhaustive numerical simulations. Furthermore, a uniaxial tensile test and mode-I fracture experiment are conducted with particle-embedded hydrogels to validate the method. Results: Numerical results corroborate a theoretical expression of measurement error. They show that the FD error increases while the measurement error decreases for a growing estimating radius. Moreover, measurement error is linearly correlated to displacement noise. A benchmark uniaxial tensile test validates the accuracy of the proposed estimator, and the near-crack-tip measurements in a tensile fracture experiment demonstrate the estimator's capabilities near a free surface, when a material undergoes large deformation and rotation. The results of the displacement and strain data are benchmarked against kinematic data obtained using an open-source DIC software, Ncorr. Computation time for both methods is compared. Conclusions: A deformation gradient tensor estimator is developed based on a particle tracking technique and a least squares routine. Theoretical error bounds on the estimator are verified by numerical simulations, and the method's capability is confirmed by physical experiments in evaluating large deformation and rotation near a free boundary. The proposed estimator is expected to open a door towards future material tests and experimental mechanics studies, especially in large deformation and large rotation scenarios. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the electro-elastostatic fields of a multiphase concentric spherical piezoelectric ensemble under general electromechanical loading in the presence or absence of body forces and body electric charges. Each phase is assumed to be spherically isotropic and at each point the poling direction is radially oriented. Thanks to the robustness of spherical harmonics, in this work we will obtain the exact solutions for the three-dimensional displacement and electric potential when the elastic, piezoelectric and dielectric constants of each phase vary as power-law functions of the radial distance. [ABSTRACT FROM AUTHOR]

Low-rank tensor completion aims to recover the missing entries of the tensor from its partially observed data by using the low-rank property of the tensor. Since rank minimization is an NP-hard problem, the convex surrogate nuclear norm is usually used to replace the rank norm and has obtained promising results. However, the nuclear norm is not a tight envelope of the rank norm and usually over-penalizes large singular values. In this paper, inspired by the effectiveness of the matrix Schatten-q norm, which is a tighter approximation of rank norm when 0 < q < 1, we generalize the matrix Schatten-q norm to tensor case and propose a Unitary Transformed Tensor Schatten-q Norm (UTT-Sq) with an arbitrary unitary transform matrix. More importantly, the factor tensor norm surrogate theorem is derived. We prove large-scale UTT-Sq norm (which is nonconvex and not tractable when 0 < q < 1) is equivalent to minimizing the weighted sum formulation of multiple small-scale UTT- S q i (with different qi and qi ≥ 1). Based on this equivalence, we propose a low-rank tensor completion framework using Unitary Transformed Tensor Multi-Factor Norm (UTTMFN) penalty. The optimization problem is solved using the Alternating Direction Method of Multipliers (ADMM) with the proof of convergence. Experimental results on synthetic data, images and videos show that the proposed UTTMFN can achieve competitive results with the state-of-the-art methods for tensor completion. [ABSTRACT FROM AUTHOR]

Modern data analytics applications are increasingly characterized by exceedingly large and multidimensional data sources. This represents a challenge for traditional machine learning models, as the number of model parameters needed to process such data grows exponentially with the data dimensions, an effect known as the curse of dimensionality. Recently, tensor decomposition (TD) techniques have shown promising results in reducing the computational costs associated with large-dimensional models while achieving comparable performance. However, such tensor models are often unable to incorporate the underlying domain knowledge when compressing high-dimensional models. To this end, we introduce a novel graph-regularized tensor regression (GRTR) framework, whereby domain knowledge about intramodal relations is incorporated into the model in the form of a graph Laplacian matrix. This is then used as a regularization tool to promote a physically meaningful structure within the model parameters. By virtue of tensor algebra, the proposed framework is shown to be fully interpretable, both coefficient-wise and dimension-wise. The GRTR model is validated in a multiway regression setting and compared against competing models and is shown to achieve improved performance at reduced computational costs. Detailed visualizations are provided to help readers gain an intuitive understanding of the employed tensor operations. [ABSTRACT FROM AUTHOR]

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if a Norden golden manifold has a constant real sectional curvature, the manifold is flat. For this reason, the notions of holomorphic-like sectional curvature and holomorphic-like bisectional curvature on the Norden golden manifold are investigated, but it is seen that these notions do not work on the Norden golden manifold. This shows the need for a new concept of sectional curvature. In this direction, a new notion of sectional curvature (Norden golden sectional curvature) is proposed, an example is given, and if this new sectional curvature is constant, the curvature tensor field of the Norden golden manifold is expressed in terms of the metric tensor field. Since the geometry of the submanifolds of manifolds with constant sectional curvature has nice properties, the last section of this paper examines the semi-invariant submanifolds of the Norden golden space form. [ABSTRACT FROM AUTHOR]

We prove a theorem on scalar‐valued functions of tensors, where "scalar" refers to absolute scalars as well as relative scalars of weight w. The present work thereby generalizes an identity referred to earlier by Rosenfeld in his publication "On the energy–momentum tensor". The theorem provides a (1,1)‐tensor identity which can be regarded as the tensor analogue of the identity following from Euler's theorem on homogeneous functions. The remarkably simple identity is independent of any internal symmetries of the constituent tensors, providing a powerful tool for deriving relations between field‐theoretical expressions and physical quantities. We apply the identity especially for analyzing the metric and canonical energy–momentum tensors of matter and gravity and the relation between them. Moreover, we present a generalized Einstein field equation for the arbitrary version of vacuum space–time dynamics—including torsion and non‐metricity. The identity allows to formulate an equivalent representation of this equation. Thereby the conjecture of a zero‐energy universe is confirmed. [ABSTRACT FROM AUTHOR]

We study an inflation model driven by a triplet of antisymmetric tensor fields, with minimal and nonminimal couplings to gravity. First, we show that the presence of a triplet of antisymmetric tensor fields can provide inherent background isotropy in the stress–energy tensor contrary to the past studies using an antisymmetric tensor field. Inflation is supported in the presence of non-minimal couplings with gravity. We perform the slow roll analysis and also analyse perturbations to the antisymmetric tensor field as well as the tensor modes of perturbed metric. The speed of gravitational waves manifested from the tensor perturbations is tuned to c. We also study the evolution of the gravitational waves, calculate their power spectrum and tensor spectral index. [ABSTRACT FROM AUTHOR]

On a (pseudo-)Riemannian manifold, we consider an operator associated to a vector field and to an affine connection, which extends, in a certain way, the Hessian of a function, study its properties and point out its relation with statistical structures and gradient Ricci solitons. In particular, we provide the necessary and sufficient condition for it to be a Codazzi tensor field, and moreover, to give rise to a statistical structure. [ABSTRACT FROM AUTHOR]

In this work, we have characterized the frame bundle F M admitting metallic structures on almost quadratic ϕ -manifolds ϕ 2 = p ϕ + q I − q η ⊗ ζ , where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ -structure to the metallic structure on F M are constructed. We also prove the existence of a metallic structure on F M with the aid of the J ˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J ˜ on F M . Finally, we construct an example of it to finish. [ABSTRACT FROM AUTHOR]

We analyse a class of stochastic advection problems by conditionally averaging the passive tracer equation with respect to a given flow state. In doing so, we obtain expressions for the turbulent diffusivity as a function of the flow statistics spectrum. When flow statistics are given by a continuous-time Markov process with a finite state space, calculations are amenable to analytic treatment. When the flow statistics are more complex, we show how to approximate turbulent fluxes as hierarchies of finite state space continuous-time Markov processes. The ensemble average turbulent flux is expressed as a linear operator that acts on the ensemble average of the tracer. We recover the classical estimate of turbulent flux as a diffusivity tensor, the components of which are the integrated autocorrelation of the velocity field in the limit that the operator becomes local in space and time. [ABSTRACT FROM AUTHOR]

In the present paper, a Finsler space Fn whose Cartan's fourth curvature tensor Rijkh satisfies R Rijkhi/l/m/n/s Cemns Rijkh + demns (δikgjh - δihgjk), Rijkh ≠ 0 where, h-covariant derivative of fourth order ( l/m/n/s is Cartan's third kind covariant differential operator), with respect to xl, xm, xn and xs successively, Clmns and dlmns are non- zero covariant vector field and covariant tensor field of third order, respectively, is introduced and such space is called as Rh generalized fourecurrent Finsler space and we denote them briefly by Rh -G-FR Fn, we obtained some generalized fourecurrent space. Also we introduced Ricci generalized fourecurrent space. [ABSTRACT FROM AUTHOR]

This paper studies the concircular, projective and conharmonic curvature tensors on 4-dimensional Lorentzian manifolds known as space-times. We obtain some properties of these tensor fields by relating the known holonomy algebras for Lorentz signature (+,+,+,-). For the space-times admitting special vector fields, such as parallel and recurrent vector fields, some theorems are proved. The eigenbivector structure of the investigated tensor fields is also examined in these spaces. These results obtained by considering the holonomy theory are associated with the algebraic classification of the Riemann curvature and Ricci tensors for Lorentz signature, and various examples related to the study are also given. [ABSTRACT FROM AUTHOR]

We study an inverse problem involving the unique recovery of lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of a boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calderón-type inverse problems. The uniqueness result and the inversion formula we prove for generalized MRT could be of independent interest and we expect it to be applicable to other inverse problems for higher order operators involving tensor perturbations. [ABSTRACT FROM AUTHOR]

The existence of dark matter is explained by a new, massive, neutral, non-symmetric, rank-2 tensor gauge boson ( Z μ ν -boson). The Z μ ν -boson can be predicted by the tensor gauge boson extension of the Electro Weak (EW) theory, proposed by Savvidy (Phys Lett B 625:341, 2005). The non-symmetric rank-2 tensor Z μ ν can be decomposed into a symmetric ( Z (μ ν) ) and anti-symmetric ( Z [ μ ν ] ) part. Based on the non-Lagrangian formulation for the free sector of the R 2 -theory proposed recently by Criado et al. (Phys Rev D 102:125031, arXiv:2010.02224, 2020), our massive anti-symmetric tensor field Z [ μ ν ] corresponds to the massive symmetric spinor field Z α β γ δ in the (2,0) irrep. For the massive Z α β γ δ with the Z 2 -symmetric Higgs portal couplings to a Standard Model (SM) particle, we compute the self-annihilation cross-section of the Z α β γ δ dark matter and calculate its relic abundance. We also study the SM-SM particle scattering due to the exchange of the massive- Z (μ ν) symmetric field at a high energy scale. This proposition may have far reaching applications in astrophysics and cosmology. [ABSTRACT FROM AUTHOR]

We characterise all linear maps A : R n × n → R n × n such that, for 1 ≤ p < n , P L p ∗ (R n) ≤ c ( A [ P ] L p ∗ (R n) + Curl P L p (R n) ) holds for all compactly supported P ∈ C c ∞ (R n ; R n × n) , where Curl P displays the matrix curl. Being applicable to incompatible, that is, non-gradient matrix fields as well, such inequalities generalise the usual Korn-type inequalities used e.g. in linear elasticity. Different from previous contributions, the results gathered in this paper are applicable to all dimensions and optimal. This particularly necessitates the distinction of different combinations between the ellipticities of A , the integrability p and the underlying space dimensions n, especially requiring a finer analysis in the two-dimensional situation. [ABSTRACT FROM AUTHOR]

In the paper, we analyze problems of elasticity tensor identification for an elastic body with incorporated thin elastic and rigid inclusions in a non-coercive case. The inclusions are assumed to be delaminated from the surrounding elastic body, thus forming interfacial cracks. We consider inequality-type boundary conditions at the crack faces with unknown set of a contact to provide a mutual non-penetration between the crack faces. The considered problems are characterized by unknown displacement field and elasticity tensor. A formulation of identification problems includes an additional information, which can be found from a measurement. A solution existence of these problems is proved. [ABSTRACT FROM AUTHOR]