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The calculation of the demagnetization field is crucial in various disciplines, including magnetic resonance imaging and micromagnetics. A standard method involves discretizing the spatial domain into finite difference cells and using demagnetization tensors to compute the field. Different demagnetization tensors can result in contributions from adjacent cells that do not approach zero, nor do their differences, even as the cell size decreases. This work demonstrates that in three-dimensional space, a specific set of magnetization tensors produces the same total demagnetization field as the Cauchy principal value when the cell size approaches zero. Additionally, we provide a lower bound for the convergence speed, validated through numerical experiments. [ABSTRACT FROM AUTHOR]

The symmetric rank problem of 2 × 2 × ⋯ × 2 symmetric tensors are related to Waring's problem of binary forms. In this paper, we characterize the symmetric rank of 2 × 2 × ⋯ × 2 symmetric tensors when the symmetric rank is 1 and 2 in arbitrary characteristic (either zero or strictly larger than the order of a tensor). Moreover, we characterize the symmetric rank for 2 × 2 × 2 symmetric tensors over the fields where the characteristic is zero or larger than three or F 2 or F 3 . [ABSTRACT FROM AUTHOR]

This paper investigates the use of the magnetic gradient tensor and its eigenvectors to derive expressions for estimating the direction of magnetisation of a uniformly magnetised right circular vertical cylinder. Expressions for the gradient tensor, the eigenvalues and eigenvectors, the normalised source strength (NSS), and the direction of magnetisation on the axis of a vertical cylinder are shown to be identical in form to those previously derived for a magnetised sphere or dipole. However, the off-axis gradient tensor field of a vertical cylinder is significantly different to that of a magnetic sphere or dipole particularly at low observation heights. These differences lessen with increasing observation height and in more compact pipes. The normalised source strength displays a directional asymmetry in the eigenvector field of the gradient tensor above a pipe which is related to the direction of magnetisation. This azimuthal asymmetry in the NSS results exclusively from the horizontal components of magnetisation, which are perpendicular to the longitudinal face of the vertical cylinder. In contrast, the vertical component of magnetisation is normal to the circular planar face and produces radial symmetry in the NSS. Additionally, the paper details some unusual properties of the NSS which are attributable to non-dipole field components in the gradient tensor field which cause pairs of eigenvectors to interchange directions when ordered by their eigenvalues, as is done when calculating the NSS. This results in spatial discontinuities in the eigenvector fields arising from crossovers in the ordered eigenvalue surfaces which vary with both magnetisation direction and observation height. This phenomenon is not present in a dipole field and thus provides direct evidence of non-dipole field components. Understanding this complexity in the eigenvector field is essential to designing methodologies which can successfully estimate the direction of magnetisation over a uniformly magnetised cylinder. One proposed method based on the peak NSS, uses airborne gradient tensor data to accurately estimate magnetisation direction over a reversely magnetised pipe-like body in the Diavik diamond field of north-western Canada. The results are verified using a full tensor inversion. [ABSTRACT FROM AUTHOR]

Within the scope of a Bianchi type-I (BI) cosmological model, we study the interacting system of spinor and electromagnetic fields and its role in the evolution of the Universe. In some earlier studies, it was found that in the case of a pure spinor field, the presence of nontrivial nondiagonal components of energy–momentum tensor (EMT) leads to some severe restrictions both on the space–time geometry and/or spinor field itself, whereas in the case of electromagnetic field with induced nonlinearity, such components impose severe restrictions on metric functions and the components of the vector potential. It is shown that in the case of interacting spinor and electromagnetic fields, the restrictions are not as severe as in the other cases and in this case, a nonlinear and massive spinor field with different components of vector potential can survive in a general Bianchi type-I space–time. [ABSTRACT FROM AUTHOR]

Full‐tensor magnetic gradiometry data have been collected commercially for the last few years. However, to date, there is still no clarity on how to compare these data to scalar total field surveys. Some users display the vertical gradient of the vertical component (Bzz) and compare that to a first vertical derivative of total field with the caveat that ‘they are similar’. Others compute the length of the measured vector and call that total field. We establish the basic formulas to calculate total field from the tensor components and demonstrate this with a real data example from Thompson, Manitoba, Canada. Another key question is whether full‐tensor interpolation is required to obtain total field from tensor data. We compare the results from using a commercial full‐tensor interpolation algorithm with standard minimum curvature of the tensor components individually and with another open‐source code that uses a radial basis function interpolator on the individual tensor components. All three applications produced a total field grid of superior quality to that calculated from a scalar total field survey available for the area of study. [ABSTRACT FROM AUTHOR]

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the "Swampland" program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not. [ABSTRACT FROM AUTHOR]

Given a conformal vector field X defined on an n -dimensional Riemannian manifold (N n , g) , naturally associated to X are the conformal factor σ , a smooth function defined on N n , and a skew symmetric (1 , 1) tensor field Ω , called the associated tensor, that is defined using the 1 -form dual to X. In this article, we prove two results. In the first result, we show that if an n -dimensional compact and connected Riemannian manifold (N n , g) , n > 1 , of positive Ricci curvature admits a nontrivial (non-Killing) conformal vector field X with conformal factor σ such that its Ricci operator R c and scalar curvature τ satisfy R c (X) = − (n − 1) ∇ σ and X (τ) = 2 σ (n (n − 1) c − τ) for a constant c , necessarily c > 0 and (N n , g) is isometric to the sphere S c n of constant curvature c. The converse is also shown to be true. In the second result, it is shown that an n -dimensional complete and connected Riemannian manifold (N n , g) , n > 1 , admits a nontrivial conformal vector field X with conformal factor σ and associated tensor Ω satisfying R c (X) = − d i v Ω and Ω (X) = 0 , if and only if (N n , g) is isometric to the Euclidean space (E n , ⟨ , ⟩) . [ABSTRACT FROM AUTHOR]

In this article, we find the different sufficient conditions for a compact minimal hypersurface M of the unit sphere S n + 1 , n ∈ Z + to be the Clifford hypersurface S ℓ (ℓ n) × S m (m n) , where ℓ , m ∈ Z + , ℓ + m = n or the sphere S n . This classification is achieved by applying constraints to the tangent and normal components of the immersion. [ABSTRACT FROM AUTHOR]

This paper considers the possibility that abstract thinking and advanced synthesis skills might encourage extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic geometric structures of current physical theories: vector fields, 1 -form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission. [ABSTRACT FROM AUTHOR]

The “field tensor” is the tensor product of the electric fields of the interacting waves during a sum‐ or difference‐frequency generation nonlinear optical interaction. It is therefore a tensor describing light interacting with matter, the latter being characterized by the “electric susceptibility tensor.” The contracted product of these two tensors of equal rank gives the light‐matter interaction energy, whether or not propagation occurs. This notion having been explicitly or implicitly present from the early pioneering studies in nonlinear optics, its practical use has led to original developments in many highly topical theoretical or experimental situations, at the microscopic as well macroscopic level throughout a variety of coherent or non‐coherent processes. The aim of this review article is to rigorously explain the field tensor formalism in the context of tensor algebra and nonlinear optics in terms of a general time‐space multi‐convolutional development, using spherical tensors, with components expressed in the frame of a common basis set of irreducible tensors, or Cartesian tensors. A wide variety of media are considered, including biological tissues and their imaging, artificially engineered by various combinations of optical and static electric fields, with the two extremes of all‐optical and purely electric poling, and also bulk single crystals. [ABSTRACT FROM AUTHOR]

Gauge theories are powerful theoretical physics tools that allow complex phenomena to be reduced to simple principles and are used in both high-energy and condensed matter physics. In the latter context, gauge theories are becoming increasingly popular for capturing the intricate spin correlations in spin liquids, exotic states of matter in which the dynamics of quantum spins never ceases, even at absolute zero temperature. We consider a spin system on a three-dimensional pyrochlore lattice where emergent gauge fields not only describe the spin liquid behavior at zero temperature but crucially determine the system's temperature evolution, with distinct gauge fields giving rise to different spin liquid phases in separate temperature regimes. Focusing first on classical spins, in an intermediate temperature regime, the system shows an unusual coexistence of emergent vector and tensor gauge fields where the former is known from classical spin ice systems while the latter has been associated with fractonic quasiparticles, a peculiar type of excitation with restricted mobility. Upon cooling, the system transitions into a low-temperature phase where an entropic selection mechanism depopulates the degrees of freedom associated with the tensor gauge field, rendering the system spin-ice-like. We further provide numerical evidence that in the corresponding quantum model, a spin liquid with coexisting vector and tensor gauge fields has a finite window of stability in the parameter space of spin interactions down to zero temperature. Finally, we discuss the relevance of our findings for non-Kramers magnetic pyrochlore materials. [ABSTRACT FROM AUTHOR]

We develop an abstract theory of flows of geometric 퐻-structures, i.e., flows of tensor fields defining 퐻-reductions of the frame bundle, for a closed and connected subgroup H ⊂ SO ⁢ ( n ) H\subset\mathrm{SO}(n) , on any connected and oriented 푛-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of 퐻-structures, by way of the natural infinitesimal action of GL ⁢ ( n , R ) \mathrm{GL}(n,\mathbb{R}) on tensors combined with various bundle decompositions induced by 퐻-structures. We compute evolution equations for the intrinsic torsion under general flows of 퐻-structures and, as applications, we obtain general Bianchi-type identities for 퐻-structures, and, for closed manifolds, a general first variation formula for the L 2 L^{2} -Dirichlet energy functional ℰ on the space of 퐻-structures. We then specialise the theory to the negative gradient flow of ℰ over isometric 퐻-structures, i.e., their harmonic flow. The core result is an almost-monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulas by Chen–Struwe [M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 3, 485–502; Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 1, 83–103] for the harmonic map heat flow. This yields an 휀-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the L ∞ L^{\infty} -norm of initial torsion, in the spirit of Chen–Ding [Y. M. Chen and W. Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), 3, 567–578]. Moreover, below a certain energy level, the absence of a torsion-free isometric 퐻-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat 푛-tori, so long as the set [ S n , SO ⁢ ( n ) / H ] [\mathbb{S}^{n},\mathrm{SO}(n)/H] of homotopy classes of maps S n → SO ⁢ ( n ) / H \mathbb{S}^{n}\to\mathrm{SO}(n)/H contains more than one element and the universal cover of SO ⁢ ( n ) / H \mathrm{SO}(n)/H is a sphere, e.g. when n = 7 n=7 and H = G 2 H=\mathrm{G}_{2} , or n = 8 n=8 and H = Spin ⁢ ( 7 ) H=\mathrm{Spin}(7) . [ABSTRACT FROM AUTHOR]

In this paper, we first consider the 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M. Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold. [ABSTRACT FROM AUTHOR]

We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold (M, g) which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold (M n , g) is of order n and thereby close a small gap in a paper by Cheeger and Tian. [ABSTRACT FROM AUTHOR]

Arbitrary connections on a generic Hopf algebra H are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an H -valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on S L q (2). [ABSTRACT FROM AUTHOR]

We show we can derive the so called polar representation of 2D symmetric second and fourth order real tensors essentially just relying on the spectral theorem for unitary tensors in the complex field. The use of a coordinate-free approach allows us to clearly and methodically detect rotation-invariant quantities and to readily (and directly) deduce the representation theorems. [ABSTRACT FROM AUTHOR]

It is challenging to automatically explore an unknown 3D environment with a robot only equipped with depth sensors due to the limited field of view. We introduce THP, a tensor field-based framework for efficient environment exploration which can better utilize the encoded depth information through the geometric characteristics of tensor fields. Specifically, a corresponding tensor field is constructed incrementally and guides the robot to formulate optimal global exploration paths and a collision-free local movement strategy. Degenerate points generated during the exploration are adopted as anchors to formulate a hierarchical TSP for global path optimization. This novel strategy can help the robot avoid long-distance round trips more effectively while maintaining scanning completeness. Furthermore, the tensor field also enables a local movement strategy to avoid collision based on particle advection. As a result, the framework can eliminate massive, time-consuming recalculations of local movement paths. We have experimentally evaluate our method with a ground robot in 8 complex indoor scenes. Our method can on average achieve 14% better exploration efficiency and 21% better exploration completeness than state-of-the-art alternatives using LiDAR scans. Moreover, compared to similar methods, our method makes path decisions 39% faster due to our hierarchical exploration strategy. [ABSTRACT FROM AUTHOR]

This paper concentrates on perturbation theory concerning the tensor T-eigenvalues within the framework of tensor-tensor multiplication. Notably, it serves as a cornerstone for the extension of semidefinite programming into the domain of tensor fields, referred to as T-semidefinite programming. The analytical perturbation analysis delves into the sensitivity of T-eigenvalues for third-order tensors with square frontal slices, marking the first main part of this study. Three classical results from the matrix domain into the tensor domain are extended. Firstly, this paper presents the Gershgorin disc theorem for tensors, demonstrating the confinement of all T-eigenvalues within a union of Gershgorin discs. Afterward, generalizations of the Bauer-Fike theorem are provided, each applicable to different cases involving tensors, including those that are F-diagonalizable and those that are not. Lastly, the Kahan theorem is presented, addressing the perturbation of a Hermite tensor by any tensors. Additionally, the analysis establishes connections between the T-eigenvalue problem and various optimization problems. The second main part of the paper focuses on tensor pseudospectra theory, presenting four equivalent definitions to characterize tensor ε -pseudospectra. Accompanied by a thorough analysis of their properties and illustrative visualizations, this section also explores the application of tensor ε -pseudospectra in identifying more T-positive definite tensors. [ABSTRACT FROM AUTHOR]

In this paper, we analyze energy density inhomogeneities for charged fluid configuration in the background of f (T) theory and recognize its prime features as computed in GR. The dynamical equations are composed employing Bianchi identities for the standard, f (T) extra terms, and energy-momentum tensor for the electromagnetic field. We evaluate various mathematical models of dissipative and anisotropic fluid distributions in-plane symmetry under f (T) gravity. To proceed with the investigation, we design the f (T) field equations, kinematical quantities, and mass function. We analyzed dynamical variables and Ellis equations in terms of our considered theory. To examine the associated inhomogeneity factors, specific scenarios are illustrated alongside and without dissipation. Within a non-radiating situation, we analyze dust and isotropic and anisotropic matter in the state of electric charge. We examine the inhomogeneity factor of a dissipative fluid via a charged dust haze. We derive that the electromagnetic field fosters matter inhomogeneity, but additional curvature factors make the entire structure more homogenous as time passes. [ABSTRACT FROM AUTHOR]

We investigate the space-time decay rates of solutions to the 3D Cauchy problem of the compressible Oldroyd-B system with diffusive properties and without viscous dissipation. The main novelties of this paper involve two aspects: On the one hand, we prove that the weighted rate of k-th order spatial derivative (where 0≤k≤3) of the global solution (ρ,u,η,τ) is t-3/4+k/2+γ in the weighted Lebesgue space L²γ. On the other hand, we show that the space-time decay rate of the m-th order spatial derivative (where m∈[0,2]) of the extra stress tensor of the field in L02γ is (1+t)-5/4-m/2+γ, which is faster than that00 of the velocity. The proofs are based on delicate weighted energy methods and interpolation tricks. [ABSTRACT FROM AUTHOR]

In this work we develop a numerical method for solving a type of convex graphstructured tensor optimization problem. This type of problem, which can be seen as a generalization of multimarginal optimal transport problems with graph-structured costs, appears in many applications. Examples are unbalanced optimal transport and multispecies potential mean field games, where the latter is a class of nonlinear density control problems. The method we develop is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures it is possible to derive efficient methods for computing these projections, and here we specifically consider the graph structure that occurs in multispecies potential mean field games. We also illustrate the methodology on a numerical example from this problem class. [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. [ABSTRACT FROM AUTHOR]

The total magnetization of an underground magnetic source is the vector sum of the induced magnetization and the natural remanent magnetization. The direction of the total magnetization serves as important a priori information in the inversion and processing of magnetic data. We demonstrated that the total gradient of the magnetic potential with vertical magnetization constitutes the envelope of the vertical component of the magnetic field for all directions of the Earth's field and source magnetization. The total gradient of the magnetic potential with vertical magnetization and the reduction‐to‐the‐pole field simultaneously tend to achieve maximum symmetry near the correct total magnetization direction. As a result, the total magnetization direction can be estimated by computing the correlations between the reduction‐to‐the‐pole and the total gradient of the magnetic potential with vertical magnetization. The proposed method yields accurate magnetization directions in synthetic model examples. The total gradient of the magnetic potential with vertical magnetization is less susceptible to data noise than transforms which are derived from the high‐order magnetic field derivatives or tensors. The estimation results are slightly affected by changes in the source magnetization direction. In a field example in the Weilasito region (North China), the reduction‐to‐the‐pole fields calculated using the estimated magnetization directions are well centred over the source. The proposed method obtained a more focused magnetization direction than that of a three‐dimensional magnetization vector inversion. The total gradient of the magnetic potential with vertical magnetization therefore provides a novel and accurate approach to determine the total magnetization direction from the total field anomaly in a variety of situations. [ABSTRACT FROM AUTHOR]

This paper focuses on studying the denoising problem for positive semidefinite fourth-order tensor field estimation from noisy observations. The positive semidefiniteness of the tensor is preserved by mapping the tensor to a 6-by-6 symmetric positive semidefinite matrix where its matrix rank is less than or equal to three. For denoising, we propose to use an anisotropic discrete total variation function over the tensor field as the regularization term. We propose an inexact majorized proximal alternating direction method of multipliers for such a nonconvex and nonsmooth optimization problem. We show that an ε-stationary solution of the resulting optimization problem can be found in no more than O (ε-4) iterations. The effectiveness of the proposed model and algorithm is tested using multifiber diffusion weighted imaging data, and our numerical results demonstrate that our method outperforms existing methods in terms of denoising performance. [ABSTRACT FROM AUTHOR]

In this article we characterize the range of the attenuated and non-attenuated X-ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with A-analytic maps in the sense of Bukhgeim. [ABSTRACT FROM AUTHOR]

Space plasma turbulence is inherently characterized by anisotropic fluctuations. The generalized k-th order correlation tensor of magnetic field increments allow us to separate the mixed isotropic and anisotropic structure functions from the purely anisotropic ones. In this work, we quantified the relative importance of anisotropic fluctuations in solar wind turbulence using two Alfvénic data samples gathered by the Solar Orbiter at 0.8 astronomical units. The results based on the joined statistics suggest that the anisotropic fluctuations are ubiquitous in solar wind turbulence and persist at kinetic scales. Using the R T N coordinate system, we show that their presence depends on the anisotropic sector under consideration, e.g., the R N and R T sectors exhibit enhanced anisotropy toward kinetic scales, in contrast with the T N . We then study magnetic field fluctuations parallel and perpendicular to the local mean magnetic field separately. We find that perpendicular fluctuations are representative of the global statistics, resembling the typical picture of magnetohydrodynamic turbulence, whereas parallel fluctuations exhibit a scaling law with slope ∼1 for all the joined isotropic and anisotropic components. These results are in agreement with predictions based on the critical balance phenomenology. This topic is potentially of interest for future space missions measuring kinetic and MHD scales simultaneously in a multi-spacecraft configuration. [ABSTRACT FROM AUTHOR]

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed. [ABSTRACT FROM AUTHOR]

Reconstructing a three-dimensional ocean sound speed field (SSF) from limited and noisy measurements presents an ill-posed and challenging inverse problem. Existing methods used a number of pre-specified priors (e.g., low-rank tensor and tensor neural network structures) to address this issue. However, the SSFs are often too complex to be accurately described by these pre-defined priors. While utilizing neural network-based priors trained on historical SSF data may be a viable workaround, acquiring SSF data remains a nontrivial task. This work starts with a key observation: Although natural images and SSFs admit fairly different characteristics, their denoising processes appear to share similar traits—as both remove random components from more structured signals. This observation allows us to incorporate deep denoisers trained using extensive natural images to realize zero-shot SSF reconstruction, without any extra training or network modifications. To implement this idea, an alternating direction method of multipliers (ADMM) algorithm using such a deep denoiser is proposed, which is reminiscent of the plug-and-play scheme from medical imaging. Our plug-and-play framework is tailored for SSF recovery such that the learned denoiser can be simultaneously used with other handcrafted SSF priors. Extensive numerical studies show that the new framework largely outperforms state-of-the-art baselines, especially under widely recognized challenging scenarios, e.g., when the SSF samples are taken as tensor fibers. The code is available at https://github.com/OceanSTARLab/DeepPnP. [ABSTRACT FROM AUTHOR]

The Neural Radiance Field (NeRF) exhibits excellent performance for view synthesis tasks, but it requires a large amount of memory and model parameters during three-dimensional (3D) scene reconstruction. This paper proposes a block-term tensor decomposition radiance field (BTD-RF), which is a novel approach that achieves significant model compression while preserving reconstruction quality. BTD-RF decomposes high-dimensional radiance fields into low-dimensional tensor blocks, resulting in a value 2.21 times smaller than the baseline method. Decomposing the model into low-dimensional tensor blocks allows substituting the standard multi-head attention of transformers with a lightweight multi-linear attention mechanism, employing element-wise products and sharing parameters. This significantly reduces the model complexity without compromising performance. Extensive evaluations on various datasets demonstrate that BTD-RF achieves superior image reconstruction quality compared to prior methods. Quantitative metrics and qualitative assessments confirm that BTD-RF generates images that are structurally and perceptually close to ground truth, showcasing exceptional performance despite its lightweight design. BTD-RF offers a compelling trade-off between model size and reconstruction quality for three-dimensional (3D) scene reconstruction. Its efficient design makes it suitable for resource-constrained applications while delivering high-fidelity results, paving the way for broader NeRF utilization. The code is available at https://github.com/seonbin-kim/BTDRF [ABSTRACT FROM AUTHOR]

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0, 2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes. [ABSTRACT FROM AUTHOR]

As the high-frequency analogue to field-scale earthquakes, acoustic emissions (AEs) provide a valuable complement to study rock deformation mechanisms. During the load-stepping creep experiments with CO2-saturated water injection into a basaltic sample from Carbfix site in Iceland, 8791 AE events are detected by at least one of the seven piezoelectric sensors. Here, we apply a cross-correlation-based source imaging method, called geometric-mean reverse-time migration (GmRTM) to locate those AE events. Besides the attractive picking-free feature shared with other waveform-based methods (e.g. time-reversal imaging), GmRTM is advantageous in generating high-resolution source images with reduced imaging artefacts, especially for experiments with relatively sparse receivers. In general, the imaged AE locations are found to be scattered across the sample, suggesting a complicated fracture network rather than a well-defined major shear fracture plane, in agreement with X-ray computed tomography imaging results after retrieval of samples from the deformation apparatus. Clustering the events in space and time using the nearest-neighbour approach revealed a group of 'repeaters', which are spatially co-located over an elongated period of time and likely indicate crack, or shear band growth. Furthermore, we select 2196 AE events with high signal-to-noise-ratio (SNR) and conduct moment tensor estimation using the adjoint (backpropagated) strain tensor fields at the locations of AE sources. The resulting AE locations and focal mechanisms support our previously assertion that creep of basalt at the experimental conditions is accommodated dominantly by distributed microcracking. [ABSTRACT FROM AUTHOR]

Discrimination of underground explosions from naturally occurring earthquakes and other anthropogenic sources is one of the fundamental challenges of nuclear explosion monitoring. In an operational setting, the number of events that can be thoroughly investigated by analysts is limited by available resources. The capability to rapidly screen out events that can be robustly identified as not being explosions is, therefore, of great potential benefit. Nevertheless, possible mis-classification of explosions as earthquakes currently limits the use of screening methods for verification of test-ban treaties. Moment tensors provide a physics-based classification tool for the characterization of different seismic sources and have enabled the advent of new techniques for discriminating between earthquakes and explosions. Following normalization and projection of their six-degree vectors onto the hypersphere, existing screening approaches use spherically symmetric metrics to determine whether any new moment tensor may have been an explosion. Here, we show that populations of moment tensors for both earthquakes and explosions are anisotropically distributed on the hypersphere. Distributions possessing elliptical symmetry, such as the scaled von Mises–Fisher distribution, therefore provide a better description of these populations than the existing spherically symmetric models. We describe a method that uses these elliptical distributions in combination with a Bayesian classifier to achieve successful classification rates of 99 per cent for explosions and 98 per cent for earthquakes using existing catalogues of events from the western United States. The 1983 May 5 Crowdie underground nuclear test and 2018 July 20 DAG-1 deep-borehole chemical explosion are the only two explosions out of 140 that are incorrectly classified. Application of the method to the 2006–2017 nuclear tests in the Democratic People's Republic of Korea yields 100 per cent identification rates and we provide a simple routine MTid for general usage. The approach provides a means to rapidly assess the likelihood of an event being an explosion and can be built into monitoring workflows that rely on simultaneously assessing multiple different discrimination metrics. [ABSTRACT FROM AUTHOR]

Sen's action for a p-form gauge field with self-dual field strength coupled to a spacetime metric g involves an explicit Minkowski metric and the presence of this raises questions as to whether the action is coordinate independent and whether it can be used on a general spacetime manifold. A natural generalisation of Sen's action is presented in which the Minkowski metric is replaced by a second metric on spacetime. The theory is covariant and can be formulated on any spacetime. The theory describes a physical sector, consisting of the chiral p-form gauge field coupled to the dynamical metric g, plus a shadow sector consisting of a second chiral p-form and the second metric . The fields in this shadow sector only couple to each other and have no interactions with the physical sector, so that they decouple from the physical sector. The resulting theory is covariant and can be formulated on any spacetime. Explicit expressions are found for the interactions and extensions to include interactions with other physical fields or higher-derivative field equations are given. A spacetime with two metrics has some interesting geometry and some of this is explored here and used in the construction of the interactions. The action has two diffeomorphism-like symmetries, one acting only on the physical sector and one acting only on the shadow sector, with the spacetime diffeomorphism symmetry arising as the diagonal subgroup. This allows a further generalisation in which is not a tensor field but is instead a gauge field whose transition functions involve the usual coordinate transformation together with a shadow sector gauge transformation. [ABSTRACT FROM AUTHOR]

A warped product submanifold, whose tangent bundle can be decomposed to two orthogonal distributions; invariant and pointwise slant, is called a warped product pointwise semi-slant submanifold. The objective of this paper is to classify warped product pointwise semi-slant submanifolds, isometrically immersed into a Sasakian manifold. Park [25] provided the (non)-existence of a warped product pointwise semi-slant submanifold in a Sasakian manifold such that the structure vector field is tangential to fibers. In contrast, we provide intriguing theorems on warped product pointwise semi-slant submanifolds in a Sasakian manifold in terms of the shape operator and tensor fields such that the structure vector field is tangential to a base manifold and each fiber is a pointwise slant submanifold. [ABSTRACT FROM AUTHOR]

Pretrained models in abstractive summarization enable the summarization model to generate coherent summaries flexibly, but it also introduces unfactual hallucinations when generating novel words. To circumvent this drawback, we propose A Guided Tensor Product framework for abstractive Summarization (GTPSum). GTPSum utilizes highlighted sentence guidance signal and role embedding in the form of guided tensor product representations (GTPRs) to enhance the generation capacity for short document text. The guidance signal assists the model in focusing on crucial information, while the role embedding aids GTPSum in capturing the syntactic structure relationships present in the sentences. By employing this approach, the framework can not only focus on the essential information from the source document but also grasp the inherent sentence structure, resulting in generated summaries that closely align with the original content. The experimental results demonstrate the strong performance of our approach on the CNN/Daily Mail (44.28/20.88/41.08 ROUGE-1/-2/-L) and performed well on the other three datasets. Moreover, the comparison between our model generated summaries and those generated by the baseline models through human evaluation shows that our summaries contain more critical information, have better readability, and still maintain succinctness. [ABSTRACT FROM AUTHOR]

Many‐body expansion (MBE) fragment approaches have been applied to accurately compute nuclear magnetic resonance (NMR) parameters in crystalline systems. Recent examples demonstrate that electric field gradient (EFG) tensor parameters can be accurately calculated for 14N and 17O. A key additional development is the simple molecular correction (SMC) approach, which uses two one‐body fragment (i.e., isolated molecule) calculations to adjust NMR parameter values established using 'benchmark' projector augmented‐wave (PAW) density functional theory (DFT) values. Here, we apply a SMC using the hybrid PBE0 exchange‐correlation (XC) functional to see if this can improve the accuracy of calculated 35Cl EFG tensor parameters. We selected eight organic and two inorganic crystal structures and considered 15 chlorine sites. We find that this SMC improves the accuracy of computed values for both the 35Cl quadrupolar coupling constant (CQ) and the asymmetry parameter (ηQ) by approximately 30% compared with benchmark PAW DFT values. We also assessed a SMC that offers local improvements not only in terms of the quality of the XC functional but simultaneously in the quality of the description of relativistic effects via the inclusion of spin–orbit effects. As the inorganic systems considered contain heavy atoms bonded to the chlorine atoms, we find further improvements in the accuracy of calculated 35Cl EFG tensor parameters when both a hybrid functional and spin–orbit effects are included in the SMC. On the contrary, for chlorine‐containing organics, the inclusion of spin–orbit relativistic effects using a SMC does not improve the accuracy of computed 35Cl EFG tensor parameters. [ABSTRACT FROM AUTHOR]

Coal mining in thick-hard roof areas leads to frequent high-energy seismicity and may even induce coal–rock dynamic disasters such as rockburst. It is important to explore the focal mechanism of mining-induced seismicity to understand the failure behaviour of the coal–rock mass and the characteristics of the stress field. It is helpful to propose a seismic absorption method. Taking the 6306 working face of the Dongtan coal mine as the study area, the rupture type, focal mechanism, characteristics of the local stress field, and main controlling factors of the mining-induced 22 high-energy seismic events were investigated using the probabilistic full-waveform optimization inversion and the stress field inversion. The main conclusions were as follows: (1) the rupture types of 22 seismic events were shear, tensile, compression, and mixed failure, and the corresponding number of events was 11, 4, 1, and 6, respectively; (2) the focal mechanism of the above events was eight strike-slips, twelve reverse fault slips, and two normal fault slips; and (3) the stress field using the inversion of focal mechanisms was discovered. There was no significant variation of the maximum compressive stress σ1 and minimum compressive stress σ3 directions in the seismicity concentration area, fold 1, and fold 2. The principal stresses σ1 and σ3 were the main controlling factors inducing seismicity. This study is helpful in revealing the mechanism of the mining-induced high-energy seismicity and proposing corresponding prevention measures in the future. Highlights: A possible full-waveform optimization inversion method is developed to analyse the focal mechanism of a thick-hard roof zone The main controlling factors of the induced seismic activity are investigated through the stress field inversion The stress field inversion results have excellent potential to guide the design of hydraulic fracturing in the coal mine [ABSTRACT FROM AUTHOR]

We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n-tensor heat flow problem. These methods differ with respect to the surface representation used, the geometric information required, and the treatment of the tangentiality condition. We emphasize the importance of geometric properties and their increasing influence as the tensorial degree changes from n = 0 to n ⩾ 1. A specific example is presented that illustrates how curvature drastically affects the behavior of the solution. [ABSTRACT FROM AUTHOR]

Motivated by recent discussions and the absence of exact global symmetries in UV completions of gravity we re-examine the axion quality problem (and naturalness issues more generally) using antisymmetric Kalb-Ramond (KR) fields rather than their pseudoscalar duals, as suggested by string and higher dimensional theories. Two types of axions can be identified: a model independent S-type axion dual to a two form Bμν in 4D and a T-type axion coming directly as 4D scalar Kaluza-Klein (KK) components of higher-dimensional tensor fields. For T-type axions our conclusions largely agree with earlier workers for the axion quality problem, but we also reconcile why T-type axions can couple to matter localized on 3-branes with Planck suppressed strength even when the axion decay constants are of order the KK scale. For S-type axions, we review the duality between form fields and massive scalars and show how duality impacts naturalness arguments about the UV sensitivity of the scalar potential. In particular UV contributions on the KR side suppress contributions on the scalar side by powers of m/M with m the axion mass and M the UV scale. We re-examine how the axion quality problem is formulated on the dual side and compare to recent treatments. We study how axion quality is affected by the ubiquity of p-form gauge potentials (for both p = 2 and p = 3) in string vacua and identify two criteria that can potentially lead to a problem. We also show why most fields do not satisfy these criteria, but when they do the existence of multiple fields also provides mechanisms for resolving it. We conclude that the quality problem is easily evaded. [ABSTRACT FROM AUTHOR]

We classify all higher-order generalised Einstein-Maxwell Lagrangians that include terms linear in the curvature tensor and quadratic in the derivatives of the electromagnetic field strength tensor. Using redundancies due to the Bianchi identities, dimensionally dependent identities and boundary terms, we show that a general Lagrangian of this form can always be reduced to a linear combination of only 21 terms, with coefficients that are arbitrary functions of the two scalar invariants derived from the field strength. We give an explicit choice of basis where these 21 terms include 3 terms linear in the Riemann tensor and 18 terms quadratic in the derivatives of the field strength. [ABSTRACT FROM AUTHOR]

General Relativity (GR) will soon celebrate its 110th birthday, holding up against all experimental enquiry. Nonetheless, unification theories attempting to quantize gravity, such as string theory, are gaining footing. These hypothesize additional scalar, vector, and tensor long-range fields that couple to matter (Will, 2014), introducing violations to GR. Although such violations have never been detected, it is likely that GR will not be the ultimate theory of gravity. What is certain is that gravity tests are alive and well, pushing the validity of GR to new scales and accuracies, or -potentially- suggesting alternative routes for new physics. Building upon the legacy of Voyager and Pioneer missions, which demonstrated the capability to survive in the outer reaches of the solar system, the Interstellar Probe mission concept (McNutt et al., 2022) aims to characterise our heliosphere through state-of-the-art instrumentation, opening new frontiers also for GR testing. In this work, we investigate the possibility of constraining the Nordtvedt parameter η and the mass of the graviton via the Compton wavelength λ C , by simulating the processing of 10 years of radiometric data from the Interstellar Probe. Station calibration and clock synchronisation, as well as limiting spacecraft precession manoeuvres are highlighted as key strategies for obtaining high-quality estimates. In the most favourable scenario, η can be constrained to less than 1.5 · 10 - 5 , reducing the uncertainty obtained via Lunar Laser Ranging (Hofmann and Müller, 2018), and a lower bound of 1.4 · 10 14 km is set for λ C , improving the estimates obtained from planetary ephemerides (Bernus et al., 2020) and gravitational wave detection (Abbott et al., Jun 2021). Extending ranging measurement acquisition to 20 years improves the results tenfold. This experiment interrogates fundamental physics from a unique dynamical setting, investigating possible violations of the Equivalence Principle (EP) underlying GR. [ABSTRACT FROM AUTHOR]

In this work, we study the topological phase transitions of a Kitaev chain generalized by the addition of nearest-neighbor Coulomb interaction. We show the presence of a robust topological phase as a function of the interaction strength and of the on-site energy with associated non-zero energy Majorana states localized at the chain edges. We provide an effective mean-field model that allows for the self-consistent computation of the mean value of the local particle number operator, and we also perform Density Matrix Renormalization Group numerical simulations based on a tensor network approach. We find that the two methods show a good agreement in reporting the phase transition between trivial and topological superconductivity. Temperature robustness within a physically relevant threshold has also been demonstrated. These findings shed light on an entire class of topological interacting one-dimensional systems in which the effects of residual Coulomb interactions play a relevant role. [ABSTRACT FROM AUTHOR]

We investigate thick brane solutions in the Horndeski gravity. In this setup, we found analytical solutions, applying the first-order formalism to two scalar fields where the first field comes from the nonminimal scalar-tensor coupling and the second is due to the matter contribution sector. With these analytical solutions, we evaluate the symmetric thick brane solutions in Horndeski gravity with four-dimensional geometry. In such a setup, we evaluate the gravity fluctuations to find "almost massless modes," for any values of the Horndeski parameters. These modes were used to compute the corrections to the Newtonian potential and evaluate the limit four-dimensional gravity. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study some curvature properties of generalized B -curvature tensor on Lorentzian β -Kenmotsu manifold. Here first, we describe certain vanishing properties of generalized B -curvature tensor on Lorentzian β -Kenmotsu manifold and obtained several interesting results. Next, we formulate ϕ − B semi-symmetric condition on Lorentzian β -Kenmotsu manifold. Again, we discussed the generalized B pseudo-symmetric condition on Lorentzian β -Kenmotsu manifold. We also characterized generalized B ϕ -recurrent Lorentzian β -Kenmotsu manifold. Further, we deal with Lorentzian β -Kenmotsu manifold satisfying B ((U , V) ⋅ Q) X = 0 condition. [ABSTRACT FROM AUTHOR]

By means of ϵ and large N expansions, we study generalizations of the O(N) model where the fundamental fields are tensors of rank r rather than vectors, and where the global symmetry (up to additional discrete symmetries and quotients) is O(N)r, focusing on the cases r ≤ 5. Owing to the distinct ways of performing index contractions, these theories contain multiple quartic operators, which mix under the RG flow. At all large N fixed points, melonic operators are absent and the leading Feynman diagrams are bubble diagrams, so that all perturbative fixed points can be readily matched to full large N solutions obtained from Hubbard-Stratonovich transformations. The family of fixed points we uncover extend to arbitrary higher values of r, and as their number grows superexponentially with r, these theories offer a vast generalization of the critical O(N) model. We also study sextic O(N)r theories, whose large N limits are obscured by the fact that the dominant Feynman diagrams are not restricted to melonic or bubble diagrams. For these theories the large N dynamics differ qualitatively across different values of r, and we demonstrate that the RG flows possess a numerous and diverse set of perturbative fixed points beginning at rank four. [ABSTRACT FROM AUTHOR]

Recently, a "no inner (Cauchy) horizon theorem" for static black holes with non-trivial scalar hairs has been proved in Einstein–Maxwell–scalar theories and also in Einstein–Maxwell–Horndeski theories with the non-minimal coupling of a charged (complex) scalar field to Einstein tensor. In this paper, we study an extension of the theorem to the static black holes in Einstein–Maxwell–Gauss–Bonnet-scalar theories, or simply, charged Gauss–Bonnet (GB) black holes. We find that no inner horizon with charged scalar hairs is allowed for the planar ( k = 0 ) black holes, as in the case without GB term. On the other hand, for the non-planar ( k = ± 1 ) black holes, we find that the haired inner horizon can not be excluded due to GB effect generally, though we can not find a simple condition for its existence. As some explicit examples of the theorem, we study numerical GB black hole solutions with charged scalar hairs and Cauchy horizons in asymptotically anti-de Sitter space, and find good agreements with the theorem. Additionally, in an Appendix, we prove a "no-go theorem" for charged de Sitter black holes (with or without GB terms) with charged scalar hairs in arbitrary dimensions. [ABSTRACT FROM AUTHOR]

The conformal version of the antisymmetric second-order tensor field in four spacetime dimensions does not have gauge invariance extensively discussed in the literature for more than half a century. Our first observation is that, when coupled to fermions, only the conformal version provides renormalizability of the theory at the one-loop level. General considerations are supported by the derivation of one-loop divergences in the fermionic sector, indicating good chances for asymptotic freedom. The arguments concerning one-loop renormalizability remain valid in the presence of self-interactions and the masses for both fermion and antisymmetric tensor fields. In the flat spacetime limit, even regardless the conformal symmetry has gone, there is an expectation to meet renormalizability in all loop orders. [ABSTRACT FROM AUTHOR]

Tensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data T=RQTΛQ$$ \boldsymbol{T}=\boldsymbol{R}{\boldsymbol{Q}}^T\kern0.00em \boldsymbol{\Lambda} \boldsymbol{Q} $$, followed by an individual interpolation of the two rotation tensors R$$ \boldsymbol{R} $$ and Q$$ \boldsymbol{Q} $$ and the positive definite diagonal eigenvalue tensor Λ$$ \boldsymbol{\Lambda} $$ resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group 핊핆(3), either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, R$$ \boldsymbol{R} $$ and Q$$ \boldsymbol{Q} $$ remain orthogonal tensors and Λ$$ \boldsymbol{\Lambda} $$ remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application. [ABSTRACT FROM AUTHOR]

Enhanced tensor field theories (eTFTs) have dominant graphs that differ from the melonic diagrams of conventional tensor field theories. They therefore describe pertinent candidates to escape the so-called branched polymer phase, the universal geometry found for tensor models. For generic order d of the tensor field, we compute the perturbative β -functions at one-loop of two just-renormalizable quartic eTFT coined by + or ×, depending on their vertex weights. The models + has two quartic coupling constants (λ , λ +) , and two 2-point couplings (mass, Za). Meanwhile, the model × has two quartic coupling constants (λ , λ ×) and three 2-point couplings (mass, Za, Z 2 a ). At all orders, both models have a constant wave function renormalization: Z = 1 and therefore no anomalous dimension. Despite such peculiar behavior, both models acquire nontrivial radiative corrections for the coupling constants. The RG flow of the model + exhibits a particular asymptotic safety: λ + is marginal without corrections thus is a fixed point of arbitrary constant value. All remaining couplings determine relevant directions and get suppressed in the UV. Concerning the model ×, λ × is marginal and again a fixed point (arbitrary constant value), λ, µ and Za are all relevant couplings and flow to 0. Meanwhile Z 2 a is a marginal coupling and becomes a linear function of the time scale. This model can neither be called asymptotically safe or free. [ABSTRACT FROM AUTHOR]