Tensor Space Research Articles

Extension criterion involving the middle eigenvalue of the strain tensor on local strong solutions to the 3D Navier–Stokes equations

In this article, we prove an extension criterion for a local strong solution to the 3D Navier–Stokes equations that only require control of the positive part of middle eigenvalue of strain tensor in the critical endpoint Besov space, i.e., λ2+∈L2(0,T;Ḃ∞,∞−1). This gives a positive answer to the problem proposed by Miller [1] and improves the results by Wu [2–4]. The proof relies on the identity for enstrophy growth and Lp-norm estimate of the gradient of λ2+.

Decomposing tensor spaces via path signatures

The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

Toward Jordan decompositions for tensors

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

Boolean operators and neural networks

We compute the homogeneous ideals of varieties, in a projective space of tensors, associated to different choices of the Boolean operators that describe the decision process in small neural networks. We prove that, starting with networks with three nodes, the varieties associated to different Boolean operators are all distinct.

Graded picture invariants and polynomial invariants for mixed tensor superspaces

In this paper we consider the mixed tensor space of a Z2-graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, this set turns out to be the set of supertrace monomials, which is analogous to the result of Procesi for the general linear group. A queer supergroup analogue of these results is also obtained.

Quantum supersymmetric pairs and ıSchur duality of type AIII

We construct quantum supersymmetric pairs (U,Uı) of type AIII and elucidate their fundamental properties. An ıSchur duality between the ıquantum supergroup Uı and the Hecke algebra of type B acting on a tensor space is established, providing a super generalization of the ıSchur duality of type AIII. Additionally, we construct a (quasi) K-matrix for arbitrary parameters, which facilitates the realization of the Hecke algebra action on the tensor space.

Simultaneous measurement of six displacement gradients using Digital Shearography

Optical methods show significant promise in accurately measuring all components of the displacement gradient tensor in three-dimensional space, excluding those related to depth measurement. However, dynamic measurements using Digital Shearography face challenges due to the complexity of acquiring sufficient phase information in one capture. This paper introduces a novel Digital Shearography system with triple beams and spatial phase shift capabilities. It allows simultaneous measurement of in-plane and out-of-plane phase distributions in a single image. Three laser beams, each with distinct wavelengths, are strategically positioned both horizontally and vertically to illuminate the object under test. Utilizing beam splitting and redirection, the system achieves dual shearing directions and captures shearograms using a 3CCD camera. Phase information extracted via the spatial phase-shift method enables determination of the six measurable displacement gradients and corresponding strain components. The manuscript provides a comprehensive explanation of the system’s theory, supported by experimental results, and explores potential applications.

The (1,0) tensor and hypermultiplets in loop space

We show that the (1,0) tensor and hypermultiplet supersymmetry variations can be uplifted to loop space. Upon dimensional reduction we make contact with abelian five-dimensional super Yang-Mills, which has a nonabelian generalization that we subsequently uplift back to loop space where we conjecture a nonabelian generalization of the (1,0) supersymmetry variations and demonstrate their on-shell closure.

Mixed finite elements for Kirchhoff–Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff–Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

Strassen's rank additivity for small tensors, including tensors of rank less or equal 7

The article is concerned with the problem of additivity of the tensor rank. That is, for two independent tensors, we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we present families of pairs of small three-way tensors for which the additivity holds. For instance, over the base field C, it is the case if both tensors are of rank less or equal 7. This proves that a pair of 2×2 matrix multiplication tensors has the rank additivity property. We show also that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.

Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., ω=dλ where λ is an r−1-form) is closed (i.e., dω=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains.

Strict convexity of yield surfaces of some weakly-textured materials

Let Sym0 be the space of traceless symmetric second-order tensors. We say that a polycrystalline elastic–plastic material is weakly-textured if its yield function f:Sym0→R is the sum of a texture-independent isotropic part fiso and an anisotropic part which is linear in the relevant texture coefficients. Let c>0 and S≔f−1(c)⊂Sym0 be the yield surface of the weakly-textured material in question. We present a sufficient condition (*), namely that ∇2f(S) be positive definite for each S∈S, for a smooth yield surface S to be strictly convex in Sym0. We apply this sufficient condition to weakly-textured materials with yield functions that satisfy the following conditions: (i) the yield functions f and fiso are smooth; (ii) ∇2fiso(S) is positive definite for each S in Siso≔fiso−1(c)⊂Sym0. We prove that the yield surface S⊂Sym0 of such weakly-textured material is strictly convex if the texture coefficients in f are sufficiently small. As illustration for practical applications, by appealing to condition (*) we study the strict convexity of the yield surface pertaining to a weakly-textured orthorhombic aggregate of cubic crystallites which has a quadratic yield function of the type proposed by Hill in 1948. Moreover, we show that all 35 samples of cold-rolled and annealed low-carbon steel sheets studied by Stickels and Mould have their quadratic yield functions and corresponding yield surfaces strictly convex.

Non-convex tensorial multi-view clustering by integrating [formula omitted]-based sliced-Laplacian regularization and [formula omitted]-sparsity

Consider the recent upswing in interest around multi-view clustering procedures. Such methods aim to boost clustering efficiency by leveraging information from numerous perspectives. Much research has been devoted to tensorial representation to exploit high-order correlations underlying disparate views while preserving the local geometric structure inside each view. Our research introduces a novel multi-view clustering approach. This approach creates a 3rd-order tensor, assimilating features from all perspectives. We use the t-product in the tensor space to generate the self-representation tensor from the tensorial data. We incorporate the ℓ1-based sliced-Laplacian regularization to increase our model’s resilience and introduce a fresh column-wise sparse norm: the ℓ2,p-norm with 0<p<1. This norm displays attributes of invariance, continuity, and differentiability. We present a closed-form answer to the ℓ2,p-regularized shrinkage problem, broadening its relevance to other generalized problems. Simultaneously, we propose a tensorial arctan-function as an improved surrogate for the tensor rank. This function has proven more proficient at assessing consistency across multiple viewpoints. By integrating these two components, we formulate an effective algorithm that refines our suggested model, ensuring that the constructed sequence gravitates toward the stationary KKT point. Our team conducts extensive experiments on various datasets to evaluate our model’s effectiveness, spanning diverse situations and scales. Results from these experiments emphasize that our approach establishes a novel performance standard.

Adjacency preserving maps between tensor spaces

Let r and s be positive integers such that r⩾3. Let U1,…,Ur be vector spaces over a field F and V1,…,Vs be vector spaces over a field K such that dim⁡Ui,dim⁡Vj⩾2 for all i,j. In this paper, we characterize maps ψ:⨂i=1rUi→⨂i=1sVi that preserve adjacency in both directions, which extends Hua's fundamental theorem of geometry of rectangular matrices. We also characterize related results concerning locally full maps preserving adjacency in both directions between tensor spaces, maps preserving adjacency in both directions between tensor spaces over a field all whose nonzero endomorphisms are automorphisms, and injective continuous adjacency preserving maps on finite dimensional tensor spaces over the real field.

A generalization of Hardy’s inequality to infinite tensors

Abstract In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ {\mathfrak{C}} , and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ ℭ ⁢ x k ∥ t , 1 ≤ U ⁢ ∥ x ∥ l p k \|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k} ( k = 1 , 2 k=1,2 ), where x is a sequence, ℭ ⁢ x k {\mathfrak{C}x^{k}} is a tensor, and ∥ ⋅ ∥ t , 1 {\|\cdot\|_{t,1}} , ∥ ⋅ ∥ l p {\|\cdot\|_{l_{p}}} are the tensor and sequence norms, respectively. The constant U is independent of x, and we seek the smallest possible value of U.

On the Averaging and Closure of Fiber Orientation Tensors in Virtual Process Chains

Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space.

Technical note: Coordinate representations of important differential operations in continuum mechanics

This note is the last part in the trilogy Tensors in Continuum Mechanics. The first part was focused on the representations of m-linear functions on tensor spaces, especially on their duals and transpositions. The second part was dedicated to the memory of Prof. T. Salmi, and it was focused on the curvilinear coordinate systems in continuum mechanics (in Finnish). Because the relationship between stress and their strain conjugates has been discussed in several papers, the focus in the present third part is on a more specific area of continuum mechanics: representations of important differential operations (the gradient, divergence, rotor, and Laplace) in curvilinear coordinate systems.

Duplex Hecke algebras of type B

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type [Formula: see text] which is a [Formula: see text]-algebra associated with the Weyl group [Formula: see text] of type [Formula: see text], and symmetric groups [Formula: see text] for [Formula: see text], satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type [Formula: see text] arising from the related [Formula: see text]-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type [Formula: see text] admits natural representations on certain tensor spaces. We then establish a Levi-type [Formula: see text]-Schur–Weyl duality of type [Formula: see text], which reveals the double centralizer property between such duplex Hecke algebras and [Formula: see text]quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018)].

Tensor-Based Lyapunov Deep Neural Networks Offloading Control Strategy with Cloud-Fog-Edge Orchestration

Using DNN (Deep Neural Networks) models to obtain high Quality of Services (QoS) through the cloud has become increasingly popular nowadays. Users want to use DNN by their edges (such as smartphones) anytime and anywhere. For most small and medium-sized enterprises, cloud computing resources are limited. Temporary exhaustion of resources may cause obvious service delay. For users, if all tasks are done locally, the battery capacity of the edge is too small to support such huge computing tasks. To remove this contradiction, we propose a Tensor-based Lyapunov DNN Offloading Control (TLDOC) strategy. First, we offload DNN computational tasks to cloud-fog-edge from an overall perspective. That is, layers in DNN are considered as basic offloadable objects. Second, we provide an original tensor-based Lyapunov equation and the entire process is derived in the tensor space. Lastly, we consider more key limiting factors (e.g., cellular data and remaining edge energy) to achieve better QoS. Via above contributions, our strategy reduces service delay and energy consumption for DNN cloud-fog-edge orchestration. Our experiments include two parts - detail and comparison. The experimental detail verifies that TLDOC strategy is practical and stable. Compared experiments show that our strategy could provide better QoS than existing methods on efficiency and energy saving.