Tensor Research Articles

Statistical characterization of the collective synchrotron radiation power emitted by non-ideal magnetized plasma fluids in relativistic jets

The problem of determining the collective synchrotron radiation power emitted by non-ideal magnetized plasma fluids at kinetic equilibrium in relativistic jets is addressed. A covariant statistical kinetic approach is implemented based on a novel solution for the corresponding non-isotropic kinetic distribution function (KDF). This is expressed by a Gaussian-like solution that is consistent with relativistic magnetic moment conservation holding in collisionless magnetized plasmas and predicts tensorial equation of state and pressure anisotropy which are specific for these systems. Notably, the same equilibrium admits also a convergent integrable Chapman–Enskog series expansion around a leading-order Juttner distribution, which affords the analytical calculation of continuum fluid fields. In this reference, it is shown that the statistical average of total synchrotron power evaluated over the non-isotropic KDF differs significantly from the corresponding ensemble estimate that would be trivially obtained if the underlying velocity distribution were purely isotropic. It is pointed out that the knowledge of such a statistical discrepancy on the radiation-power curve could provide an independent framework for the characterization of the physical properties of the relativistic plasma state or of the background magnetic field that permeates these astrophysical scenarios.

Kinetic closures for unmagnetized and magnetized plasmas

Parallel and perpendicular closures with cyclotron resonance effects retained for the five-moment (density, temperature, and flow velocity) fluid equations are derived by solving the kinetic equation with the Bhatnagar–Gross–Krook operator in Fourier space. For parallel propagation, the parallel closures are reduced to those of Ji et al. [Phys. Plasmas 20, 082121 (2013)]. The closures when combined to the fluid equations reproduce the fully kinetic dispersion relation that can be directly derived from the kinetic equation. The closures for the five-moment fluid system can be utilized to derive closures for the extended fluid system, which is demonstrated by deriving closures for the ten-moment system consisting of density, flow velocity, temperature, and viscosity tensor equations.

Optical manipulation of anisotropic spherical shell particles in a dual-beam trap

In this paper, based on the generalized Lorenz-Mie theory (GLMT), expressions for the scattering coefficient and the shape factor of the beam are derived for a uniaxial anisotropic spherical shell (UASS) particle illuminated by standing laser beams. Through a comprehensive analysis involving the Maxwell stress tensor equations and conservation law of electromagnetic momentum, explicit expressions for both the transverse and axial radiation force (RF) acting upon UASS particles have been analytically derived. The current theories are shown to be valid by comparison with the existing reference. To achieve a more stable capture of UASS particles, the influence of the corresponding parameters of the particle and the dual laser beams on the capture and manipulation is investigated in detail. These investigations could provide an effective way to achieve improvements in optical tweezers and can become an encouraging approach to realize the high accuracy operation of UASS particles.

HOBi-CGSTAB and HOBi-CRSTAB methods for solving some tensor equations

HOBi-CGSTAB and HOBi-CRSTAB methods for solving some tensor equations

Preconditioned BiCGSTAB and BiCRSTAB methods for solving the Sylvester tensor equation

This paper presents a biconjugate gradient stabilized (BiCGSTAB) method and a biconjugate residual stabilized (BiCRSTAB) method for solving the Sylvester tensor equation, respectively. Preconditioned BiCGSTAB and BiCRSTAB algorithms are also developed to solve the Sylvester tensor equation. The convergence of each proposed iterative algorithm is proved. Several numerical examples are shown to illustrate the effectiveness of the proposed methods.

The new Krylov subspace methods for solving tensor equations via T-product

The new Krylov subspace methods for solving tensor equations via T-product

Iterative methods for solving tensor equations based on exponential acceleration

Iterative methods for solving tensor equations based on exponential acceleration

Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the constructions of conformal scattering theories for the tensorial wave equation such as the tensorial Fackerell-Ipser and the spin ±1 Teukolsky equations on Schwarzschild spacetime. In our strategy, we construct the conformal scattering for the tensorial Fackerell-Ipser equations which are obtained from the Maxwell equation and spin ±1 Teukolsky equations. Our method combines Penrose’s conformal compactification and the energy decay results of the tensorial fields satisfying the tensorial Fackerell-Ipser equation to prove the energy equality of the fields through the conformal boundary (resp. ) and the initial Cauchy hypersurface . We will prove the well-posedness of the Goursat problem by using a generalization of Hörmander’s results for the tensorial wave equations. By using the results for the tensorial Fackerell-Ipser equations we will establish the construction of conformal scattering for the spin ±1 Teukolsky equations.

Generalized bilateral inverses of tensors via Einstein product with applications to singular tensor equations

Generalized bilateral inverses of tensors via Einstein product with applications to singular tensor equations

Slice tensor splitting method for solving tensor equation

Firstly, we define an (r,r,⋯,r)-slice tensor, and then solve an (r,r,⋯,r)-slice tensor equation by direct method. Secondly, we propose a new splitting of the coefficient tensor to develop a slice tensor splitting method, which is different from the tensor splitting methods in [1,2]. At each iteration of this method, an (r,r,⋯,r)-slice tensor equation is solved. Finally, we prove the local convergence of the proposed method under appropriate conditions, and give numerical experiments to illustrate the effectiveness of this method.

A constitutive modeling approach to simulate time‐dependent phenomena in rubber‐like materials

AbstractIndustrial rubber materials and similar rubber‐like materials exhibit complex mechanical behavior including nonlinear stress‐strain behavior up to large strains, hysteresis, Mullins effect, permanent set, as well as time‐dependent effects, like relaxation and rate dependency. The modeling of all these phenomena is a challenging task. One constitutive model that is capable to capture many effects in the large strain regime is the so‐called model of rubber phenomenology (MORPH). However, it is not able to capture any of the mentioned time‐dependent phenomena. Thus, this contribution deals with the extension of this model to simulate relaxation and rate dependency of rubber‐like materials. Thereby, a basic modeling approach is proposed, where only a slight change in the original system of tensorial equations is necessary. This approach is further extended by applying not only a constant relaxation time, but also loading history‐depended relaxation properties. For numerical simulations, a time discretization has to be applied to the constitutive equations, which is briefly shown. Finally, the extended MORPH model is applied to two different materials: an industrial rubber material and a polyurethane based adhesive. In both cases, the extended MORPH model is adjusted to uniaxial tension test data by parameter identification procedures. It will be shown that basic time‐dependent phenomena of rubber‐like materials can be captured well with the proposed extensions.

Computing the extremal nonnegative solutions of the [formula omitted]-tensor equation with a nonnegative right-side vector

We consider the tensor equation whose coefficient tensor is a nonsingular M-tensor and whose right-side vector is nonnegative. Such a tensor equation may have a large number of nonnegative solutions. It is already known that the tensor equation has a maximal nonnegative solution and a minimal nonnegative solution (called extremal solutions collectively). However, the existing proofs do not show how the extremal solutions can be computed. The existing numerical methods can find one of the nonnegative solutions, without knowing whether the computed solution is an extremal solution. In this paper, we present new proofs for the existence of extremal solutions. Our proofs are much shorter than existing ones and more importantly they give numerical methods that can compute the extremal solutions. Linear convergence of these numerical methods is also proved under mild assumptions. Some of our discussions also allow the coefficient tensor to be a Z-tensor or allow the right-side vector to have some negative elements.

DNS-Based Turbulent Closures for Sediment Transport Using Symbolic Regression

This work aims to improve the turbulence modeling in RANS simulations for particle-laden flows. Using DNS data as reference, the errors of the model assumptions for the Reynolds stress tensor and turbulence transport equations are extracted and serve as target data for a machine learning process called SpaRTA (Sparse Regression of Turbulent Stress Anisotropy). In the present work, the algorithm is extended so that additional quantities can be taken into account and a new modeling approach is introduced, in which the models can be expressed as a scalar polynomial. The resulting corrective algebraic expressions are implemented in the RANS solver SedFoam-2.0 for cross-validation. This study shows the applicability of the SpaRTA algorithm to multi-phase flows and the relevance of incorporating sediment-related quantities to the set of features from which the models are assembled. An average improvement of ca. thirty percent on various flow quantities is achieved, compared to the standard turbulence models.

Solving the constitutive equation of dilute polymeric flows: A general Fokker–Planck approach for linear elastic dumbbell models

We present a mesoscopic numerical solver for the constitutive equation of dilute polymer solutions, as described by the Fokker–Planck equation of bead-spring dumbbell models with linear spring forces, such as Oldroyd-B, FENE-P (finitely extensible nonlinear elastic with Peterlin closure), and C-FENE-P (charged FENE-P). The configuration density function of the Fokker–Planck equation is expanded into a series of Hermite polynomials, and we show that only a second order in the expansion is needed to describe the evolution of the polymer-induced stresses. The polymer-induced stress tensor is given exactly through a discrete representation of the configuration distribution function, which is obtained from a finite set of Gauss–Hermite quadrature points. In addition, we derive the conformation tensor equation, which serves as a mathematical quality check for the method. We solve the time evolution of the extra stress tensor for homogeneous flows of dilute polyelectrolyte solutions, conceptualized by the C-FENE-P model, which is a generalization of the FENE-P dumbbell model. The results are shown to be in excellent agreement with analytical and semi-analytical reference results for simple shear and extensional flows. While in this paper the focus is on linear connector forces, we believe that the principles of the derivation are extendable to other force laws.

High-Order Robust Discrete-Time Neural Dynamics for Time-Varying Multilinear Tensor Equation With $\mathcal {M}$-Tensor

The existing discrete-time neural dynamics methods for solving the multi-linear tensor equation (MTE) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> -tensor are all derived from the continuous-time one and depend on the Euler difference formula, which can not apply to essentially discrete problems and have low solution accuracy. Moreover, these methods all focus on static problems rather than time-varying ones, and thus may have unsatisfactory performance in applications with time-varying parameters. Additionally, most of these methods fail to handle the MTE with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> -tensor under noisy conditions. To remedy these issues, a high-order robust discrete-time neural dynamics (HRDND) method with a directly discrete approach is proposed for solving the time-varying MTE (TMTE) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> -tensor in this paper. Theoretical analyses on convergence and robustness are provided to prove that the proposed HRDND method is feasible and effective. Finally, simulative experiments on four time-varying numerical examples and an application derived from the Bellman equation solved by the proposed HRDND method and other four methods are given, whose results illustrate the superiority of the proposed HRDND method.

Scalar-fermion fixed points in the ε expansion

The one-loop beta functions for systems of Ns scalars and Nf fermions interacting via a general potential are analysed as tensorial equations in 4 − ε dimensions. Two distinct bounds on combinations of invariants constructed from the couplings are derived and, subject to an assumption, are used to prove that at one-loop order the anomalous dimensions of the elementary fields are universally restricted by γϕ ⩽ frac{1}{2} Nsε and γψ ⩽ Nsε. For each root of the Yukawa beta function there is a number of roots of the quartic beta function, giving rise to the concept of ‘levels’ of fixed points in scalar-fermion theories. It is proven that if a stable fixed point exists within a certain level, then it is the only such fixed point at that level. Solving the beta function equations, both analytically and numerically, for low numbers of scalars and fermions, well-known and novel fixed points are found and their stability properties are examined. While a number of fixed points saturate one out of the two bounds, only one fixed point is found which saturates both of them.

Peeling for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the asymptotic behavior along outgoing and incoming radial geodesics, i.e. the peeling property for the tensorial Fackerell–Ipser and spin [Formula: see text] Teukolsky equations on Schwarzschild spacetime. Our method combines a conformal compactification with vector field techniques to prove the two-side estimates of the energies of tensorial fields through the future and past null infinity [Formula: see text] and the initial Cauchy hypersurface [Formula: see text] in a neighborhood of spacelike infinity [Formula: see text] far away from the horizon and future timelike infinity. Our results obtain the optimal initial data which guarantees the peeling at all orders.

Deflated and restarted Krylov subspace methods for Sylvester tensor equations

Deflated and restarted Krylov subspace methods for Sylvester tensor equations

Efficient iterative method for generalized Sylvester quaternion tensor equation

Efficient iterative method for generalized Sylvester quaternion tensor equation

Solving a system of two-sided Sylvester-like quaternion tensor equations

Solving a system of two-sided Sylvester-like quaternion tensor equations