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In the equivalent single-layer theories of anisotropic laminates, the coupling tensor B describes the relation between the in- and out-of plane behaviour of the plate. This tensor has some peculiar characteristics and in particular it is not defined, so it is normally considered as an unbounded tensor. We show in this paper a way to determine some relations between the polar invariants of B and those of the tensors describing the in- and out-of plane behaviours of the laminate, A and D , respectively. These relations constitute a set of bounds for the invariants of all the tensors of the laminate, A , B and D. It is hence shown that also the components of B must satisfy some conditions. Some peculiar cases, interesting for applications, are also considered. [ABSTRACT FROM AUTHOR]

The existing magnetic target localization methods are greatly affected by the geomagnetic field and exist approximation errors. In this paper, a two-point magnetic gradient tensor localization model is established by using the spatial relation between the magnetic target and the observation points derived from magnetic gradient tensor and tensor invariants. Based on the model, the equations relating to the position vector of magnetic target are constructed. Solving the equations, a new magnetic target localization method using only a two-point magnetic gradient tensor and no approximation errors is achieved. To accurately evaluate the localization accuracy of the method, a circular trajectory that varies in all three directions is proposed. Simulation results show that the proposed method is almost error-free in the absence of noise. After adding noise, the maximum relative error percentage is reduced by 28.4% and 2.21% compared with the single-point method and the other two-point method, respectively. Furthermore, the proposed method is not affected by the variation in the distance between two observation points. At a detection distance of 20 m, the maximum localization error is 1.86 m. In addition, the experiments also verify that the new method can avoid the influence of the geomagnetic field and the variation in the distance, and achieve high localization accuracy. The average relative error percentage in the y-direction is as low as 3.78%. [ABSTRACT FROM AUTHOR]

Airborne gravity gradient data contain additional short-wavelength information about the buried geological bodies. This study develops a fast interpretation method based on the gravity gradient data for the sources' spatial location and physical property parameters. This study analyzes the advantages of the source parameter inversion method based on tensor invariants. It proposes a normalized fast-imaging method based on tensor invariants to quickly estimate the spatial location parameters of sources through the local maximum value position of the imaging results. First, the tensor invariant characteristics and the imaging method's effect in a simple model are analyzed using a theoretical model. Second, to analyze the imaging method's application eff ect in complex model conditions, the method's applicability is quantitatively analyzed using the data added with noise, superimposed anomalies of adjacent sources, and anomalies of deep and shallow geological bodies. The theoretical model's simulation results show that the model's imaging results in this study have satisfactory performance on the spatial position estimation of the sources. Finally, the method is applied to the gravity anomaly data corresponding to the Humble salt dome. The imaging results can effectively estimate the distribution of the salt dome's horizontal and depths, verifying the practicability of the method. [ABSTRACT FROM AUTHOR]

The concept of a loading surface corresponding to creep strains is formulated similarly to how the concept of a loading surface corresponding to plastic strains is introduced. An equation for the loading surface corresponding to creep strains is obtained. It is proven that, in the absence of the material viscosity, the equation of the loading surface corresponding to creep strains coincides with the equation of the loading surface corresponding to plastic strains. [ABSTRACT FROM AUTHOR]

The visualization of symmetric second-order tensor fields in two or three dimensions is still a challenging task, particularly if global structures of the data are desired. One approach is tensor field topology which provides structures characterizing the behavior of the eigenvector fields. Another widely used approach is analyzing tensor fields by means of scalar invariants, i.e., quantities invariant with respect to changes of the coordinate system. In this case, the selection of the relevant invariants might be difficult. Thus, we propose an approach which analyzes the complete invariant part of the tensor. We define extremal points for tensor fields in a mathematically rigorous way, which form curves for two-dimensional and surfaces for three-dimensional tensor fields. We propose a way to compute extremal curves or surfaces from a suitable set of two or three invariants, respectively. We also show that commonly used sets of invariants lead to the same extremal points. Consequently, extremal points contain minima and maxima of most invariants used in tensor field analysis and they are linked to the tensor field topology by containing the degenerate points. Moreover, we show that each extremal point is an extremum or a saddle of a certain invariant. The method is demonstrated on synthetic datasets as well as on stress tensor fields from structure simulations. [ABSTRACT FROM AUTHOR]

The number of walks of k steps from the node 0 to the node λ on the McKay quiver determined by a finite group G and a G -module V is the multiplicity of the irreducible G -module G λ in the tensor power V ⊗ k , and it is also the dimension of the irreducible module labeled by λ for the centralizer algebra Z k ( G ) = End G ( V ⊗ k ) . This paper explores ways to effectively calculate that number using the character theory of G . We determine the corresponding Poincaré series. The special case λ = 0 gives the Poincaré series for the tensor invariants T ( V ) G = ⨁ k = 0 ∞ ( V ⊗ k ) G and a tensor analog of Molien's formula for polynomial invariants. When G is abelian, we show that the exponential generating function for the number of walks is a product of generalized hyperbolic functions. Many graphs (such as circulant graphs) can be viewed as McKay quivers, and the methods presented here provide efficient ways to compute the number of walks on them. [ABSTRACT FROM AUTHOR]

A strength condition is obtained that takes into account the deformation anisotropy and the effect of uniform pressure on the mechanical properties of materials. [ABSTRACT FROM AUTHOR]

In many scientific fields, empirical models are employed to facilitate computational simulations of engineering systems. For example, in fluid mechanics, empirical Reynolds stress closures enable computationally-efficient Reynolds Averaged Navier Stokes simulations. Likewise, in solid mechanics, constitutive relations between the stress and strain in a material are required in deformation analysis. Traditional methods for developing and tuning empirical models usually combine physical intuition with simple regression techniques on limited data sets. The rise of high performance computing has led to a growing availability of high fidelity simulation data. These data open up the possibility of using machine learning algorithms, such as random forests or neural networks, to develop more accurate and general empirical models. A key question when using data-driven algorithms to develop these empirical models is how domain knowledge should be incorporated into the machine learning process. This paper will specifically address physical systems that possess symmetry or invariance properties. Two different methods for teaching a machine learning model an invariance property are compared. In the first method, a basis of invariant inputs is constructed, and the machine learning model is trained upon this basis, thereby embedding the invariance into the model. In the second method, the algorithm is trained on multiple transformations of the raw input data until the model learns invariance to that transformation. Results are discussed for two case studies: one in turbulence modeling and one in crystal elasticity. It is shown that in both cases embedding the invariance property into the input features yields higher performance at significantly reduced computational training costs. [ABSTRACT FROM AUTHOR]

For a finite group G and a finite-dimensional G -module V , we prove a general result on the Poincaré series for the G -invariants in the tensor algebra T ( V ) = ⨁ k ≥ 0 V ⊗ k . We apply this result to the finite subgroups G of the 2 × 2 special unitary matrices and their natural module V of 2 × 1 column vectors. Because these subgroups are in one-to-one correspondence with the simply laced affine Dynkin diagrams by the McKay correspondence, the Poincaré series obtained are the generating functions for the number of walks on the simply laced affine Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Let G be a reductive group acting on a path algebra kQ as automorphisms. We assume that G admits a graded polynomial representation theory, and the action is polynomial. We describe the quiver Q G of the smash product algebra k Q # k [ M G ] ⁎ , where M G is the associated algebraic monoid of G . We use Q G -representations to construct coherently G -invariant modules of Q . As an application, we construct algebraic semi-invariants on the quiver representation spaces from those G -invariant modules. [ABSTRACT FROM AUTHOR]

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie-Poisson bracket of rank two. This Lie-Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented. [ABSTRACT FROM AUTHOR]

The damage of a planar, elastic and initially isotropic material is considered in the framework of a classical approach where the damaged elasticity tensor is ruled by a fourth-rank symmetric damage tensor. The analysis is completely carried on using the so-called polar method for the invariant representation of tensors in R 2 . The final elastic behavior, induced by damage, can be anisotropic: all the possible situations of elastic symmetries are considered, and for each one an analytical expression for the bounds on the invariants of the damaged elastic tensor and of the damage tensor is given. An admissible domain for the damage invariants and for the damaged elastic invariants is so provided, the convexity of these domains is also proved. [ABSTRACT FROM AUTHOR]

We introduce certain quiver analogue of the determinantal variety. We study the Kempf–Lascoux–Weyman complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen–Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equations of these varieties. When the variety has codimension one, the only irreducible polynomial function is a relative tensor invariant. As a by-product, we find some vanishing condition for the Kronecker coefficients. In the end, we make a generalization from the quiver setting to the tensor setting. [ABSTRACT FROM AUTHOR]

Highlights: [•] We define the mathematics for constructing a tensor from tensor invariants. [•] We simulate the effects of noise over the complete space of tensor shape. [•] We determine the distribution of true tensor invariants given their measures. [•] The noise sensitivity of tensor invariants varies across the space of tensor shape. [Copyright &y& Elsevier]

A gear transmission dynamic model for the gear dynamic contact loading, dynamic contact stress state and dynamic contact strain state analysis is presented. A dynamic model of the transmission with four degrees of freedom is used. The transmission is analyzed using nonlinear finite elements contact formulation, using a novel software modules developed by the author used for the generalized analysis of the geared transmissions in the environment of the open source finite elements framework CODE-ASTER/ SALOME. [ABSTRACT FROM AUTHOR]

Abstract: The problem of the search of the invariants of an anisotropic elastic tensor representing the mechanical response of a complex elastic body in a two dimensional space is addressed, in particular for a tensor that does not possess all the tensor symmetries typical of classical elasticity. The invariants of the stiffness tensor are found and all the possible types of orthotropy are discussed. [Copyright &y& Elsevier]

Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy. [ABSTRACT FROM AUTHOR]

Electron paramagnetic resonance study of a natural single crystal of natrolite was carried out at the frequency ν=36.772 GHz at room temperature. The angular dependence of the four symmetry-related spectra of Fe3+ in the three crystallographic planes was fitted to a spin Hamiltonian (S=5/2) of symmetry Ci. The rank 4 crystal field tensors at tetrahedral sites were calculated using the point-charge model to determine the principal axes orientations of their cubic and trigonal components. The analysis of zero-field splitting tensors and comparison with crystal field ones suggests that Fe3+ substitutes for Al3+ with no significant distortion of the coordination tetrahedron in natrolite. Comparison of data for several natural and synthetic crystals reveals that the 4-rank zero-field splitting tensor invariants for Fe3+ at the tetrahedral oxygen-coordinated sites are distinguishably smaller than those for Fe3+ at octahedral sites. Such comparative analysis may help to determine the substitutional sites in other crystals. [Copyright &y& Elsevier]