Projective Tensor Research Articles

Conformal bootstrap equations from the embedding space operator product expansion

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the M-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-ℓ exchange, with finite ℓ-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars S and vectors V, namely ⟨SSSV⟩.

Unsupervised Deep Tensor Network for Hyperspectral-Multispectral Image Fusion.

Fusing low-resolution (LR) hyperspectral images (HSIs) with high-resolution (HR) multispectral images (MSIs) is a significant technology to enhance the resolution of HSIs. Despite the encouraging results from deep learning (DL) in HSI-MSI fusion, there are still some issues. First, the HSI is a multidimensional signal, and the representability of current DL networks for multidimensional features has not been thoroughly investigated. Second, most DL HSI-MSI fusion networks need HR HSI ground truth for training, but it is often unavailable in reality. In this study, we integrate tensor theory with DL and propose an unsupervised deep tensor network (UDTN) for HSI-MSI fusion. We first propose a tensor filtering layer prototype and further build a coupled tensor filtering module. It jointly represents the LR HSI and HR MSI as several features revealing the principal components of spectral and spatial modes and a sharing code tensor describing the interaction among different modes. Specifically, the features on different modes are represented by the learnable filters of tensor filtering layers, the sharing code tensor is learned by a projection module, in which a co-attention is proposed to encode the LR HSI and HR MSI and then project them onto the sharing code tensor. The coupled tensor filtering module and projection module are jointly trained from the LR HSI and HR MSI in an unsupervised and end-to-end way. The latent HR HSI is inferred with the sharing code tensor, the features on spatial modes of HR MSIs, and the spectral mode of LR HSIs. Experiments on simulated and real remote-sensing datasets demonstrate the effectiveness of the proposed method.

Catalog of focal mechanism solutions for the Sichuan and Yunnan region from 2012 to 2022 using the community velocity model of Southwest China

The focal mechanism solution is one of the important focal parameters for exploring fault activity and studying regional stress distribution and it has a wide range of applications. The geological structure of the Sichuan-Yunnan region in China is complex, with frequent earthquakes and abundant historical observation data, making it one of the popular areas of concern for scholars. This study utilizes the high-precision community velocity model v2.0 of southwest China, obtained through joint inversion based on multiple data methods. The Cut-And-Paste (CAP) method was employed to fit and invert the observed waveforms of 1475 events with ML ​≥ ​3.5 in the Sichuan-Yunnan region from January 2012 to December 2022, thereby constructing a catalog of double-couple focal mechanisms. By comparing the focal mechanism inversion results of small earthquakes with those from multiple one-dimensional velocity models and conducting comparative statistical analysis on events below magnitude 4, it has been demonstrated that the model used in this study provides a better fit than one-dimensional models. This contributes to establishing the lower magnitude limit for producing deeper focal mechanism solutions. This study compares the results of larger magnitude earthquakes in the catalog with those published by the Global Centroid-Moment Tensor (GCMT) project and smaller magnitude earthquakes with the catalog released by the Institute of Earthquake Forecasting, China Earthquake Administration. These comparisons serve to validate the accuracy of the catalog results. Leveraging the high-resolution velocity model, this catalog has re-examined the historical earthquake focal mechanism catalog of the Sichuan-Yunnan region. The inversion has yielded reliable results for smaller magnitudes and a greater number of events, providing additional data and support for understanding the regional stress field, active faults, the mechanisms of large earthquake genesis, and earthquake prediction efforts. Consequently, this enhances the depth of scientific research in the Sichuan-Yunnan region.

Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

Special Geometric Objects in Generalized Riemannian Spaces

In this paper, we obtained the geometrical objects that are common in different definitions of the generalized Riemannian spaces. These objects are analogies to the Thomas projective parameter and the Weyl projective tensor. After that, we obtained some geometrical objects important for applications in physics.

Projective tensor products of approximation spaces associated with positive operators

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

Doubly warped product manifolds: Investigations through the projective curvature tensor and relativistic applications

Doubly warped product manifolds: Investigations through the projective curvature tensor and relativistic applications

On the General Covariance of Natural Laws

This paper explores the concept of general covariance in natural laws through the lens of projective geometry and tensor algebra. By introducing the notions of covariance and contravariance using intuitive examples from projections and the scalar product, we illustrate how the covariance of natural laws ensures their universality and objectivity. We also discuss the role of symmetries and conservation principles in relation to the covariant nature of physical equations, highlighting the deep interplay between the mathematical structure of physical theories and the fundamental principles of nature.

THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS

Abstract We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if $(x_n) \subset X$ and $(y_n) \subset Y$ are two weakly null sequences such that $(x_n \otimes y_n)$ converges weakly in $X \widehat {\otimes }_\pi Y$ , then $(x_n \otimes y_n)$ is also weakly null.

Five-parton scattering in QCD at two loops

We compute all helicity amplitudes for the scattering of five partons in two-loop QCD in all the relevant flavor configurations, retaining all contributing color structures. We employ tensor projection to obtain helicity amplitudes in the ’t Hooft-Veltman scheme starting from a set of primitive amplitudes. Our analytic results are expressed in terms of massless pentagon functions, and are easy to evaluate numerically. These amplitudes provide important input to investigations of soft-collinear factorization and to studies of the high-energy limit. Published by the American Physical Society 2024

On tensor projections, stress or stretch vectors and their relations to Mohr’s three circles

On tensor projections, stress or stretch vectors and their relations to Mohr’s three circles

Algebraic classification of 2+1 geometries: a new approach

We present a convenient method of algebraic classification of 2+1 spacetimes into the types I, II, D, III, N and O, without using any field equations. It is based on the 2+1 analogue of the Newman–Penrose curvature scalars ΨA of distinct boost weights, which are specific projections of the Cotton tensor onto a suitable null triad. The algebraic types are then simply determined by the gradual vanishing of such Cotton scalars, starting with those of the highest boost weight. This classification is directly related to the specific multiplicity of the Cotton-aligned null directions and to the corresponding Bel–Debever criteria. Using a bivector (that is 2-form) decomposition, we demonstrate that our method is fully equivalent to the usual Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem and determining the respective canonical Jordan form of the Cotton–York tensor. We also derive a simple synoptic algorithm of algebraic classification based on the key polynomial curvature invariants. To show the practical usefulness of our approach, we perform the classification of several explicit examples, namely the general class of Robinson–Trautman spacetimes with an aligned electromagnetic field and a cosmological constant, and other metrics of various algebraic types.

Tectonic activity in Gulf of Guinea and Sub-Sahara West Africa: A validation of Freeth (1977) using focal mechanism solutions

Fault plane solutions for a group of 104; 4.0 ≤ Mw ≤ 7.1 earthquakes between January 1979 and December 2016, extracted from the Global Centroid Moment Tensor Project catalog. Were used to investigate the regional tectonic stress regime of the Gulf of Guinea region. The idea is to validate the theory of membrane tectonics put forward by Freeth (1977)[1] in which the tectonic of the Gulf of Guinea and the sub-Sahara West Africa region were described based on Freeth (1977)[1]. The tectonic of the Gulf of Guinea and the sub-Sahara West Africa region are based on the movement of the African plate, we emphasized the use of rigorous statistical tests to decide on the quality and variability of the earthquake focal mechanisms (FMSs) utilized for the stress tensor inversion analysis. To constrain our analysis, we have applied both the Algorithm of Michael and Gauss technique in our stress tensor inversion analysis of FMS obtained from the region, and the results are found to be coherent and in good agreement with each other. Both Michael (1984)[2] and Zalohar and Vrabec (2007)[3] techniques show that the regional tectonic stress regime of the Gulf of Guinea and the sub-Sahara West Africa is extensional, which is in good agreement with the work of Freeth (1977)[1]. However, our investigation concluded that the orientation of the extensional stress regime is the same as the orientation of the movement of the African plate, which is towards the Euro-Asia plate.

Projective Tensor of Almost Kahler Manifold

The work from this search is analyze the geometrical characteristics of conharmonic tensor of projectine Almost kahler menifold. we use the projective curvature tensor's flatness properties to determine the projective tensor's Almost kahler manifold compounds. (AK-manifold). And found some components Projective of Almost Kahler. proved that the manifold M is a flat holomorphic sectional tensor of projective Almost Kahlar manifold. Prove that the manifold M has J-invariant Ricci tensor. Proved that (AK-manifold) M is kahler manifold. Finally, there exists relation between Almost Kahlar manifold (AK-manifold) and Lacally confermal Kähler menifold (LCK-manifeld) has been found.

On Almost C(α)-Manifold Satisfying Certain Curvature Conditions

This research article is about the geometry of the almost C(α)- manifold. Some important properties of the almost C(α)- manifold with respect to the W_3- curvature tensor, such as W_3-flat and W_3- semi-symmetry, are investigated. The relationship of W_3- curvature tensor with Riemann, Ricci, projective, concircular and quasi-conformal curvature tensor is discussed on the almost C(α)- manifold and many important results are obtained. In addition, W_3- pseudo symmetry and W_3- Ricci pseudo symmetry are investigated for the almost C(α)- manifold. The results obtained are interesting and give an idea about the geometry of the almost C(α)- manifold.

Unconditional bases and Daugavet renormings

We prove that the space ℓ2 – and more generally every infinite dimensional Banach space with an unconditional Schauder basis – can be renormed to have a Daugavet point. As a starting point, we introduce a new diametral notion for points of the unit sphere of Banach spaces, that complements the notion of Δ-points, but is weaker than the notion of Daugavet points. We show that this notion can be used to provide another geometric characterization of the Daugavet property, as well as to recover – and even to provide new – results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Then, we observe that the study of this notion naturally leads to new renorming ideas in the context, and combine these with previous techniques from the literature to show that Daugavet points may exist even in super-reflexive spaces.

Characterization of Almost \(\eta\) -Ricci Solitons With Respect to Schouten-van Kampen Connection on Sasakian Manifolds

In this paper, we investigate Sasakian manifolds that admit almost \(\eta\) -Ricci solitons with respect to the Schouten-van Kampen connection using certain curvature tensors. Concepts of Ricci pseudosymmetry for Sasakian manifolds admitting \(\eta\)-Ricci solitons are introduced based on the selection of specific curvature tensors such as Riemann, concircular, projective, pseudo-projective, M-projective, and W2 tensors. Subsequently, necessary conditions are established for a Sasakian manifold admitting \(\eta\)-Ricci soliton with respect to the Schouten-van Kampen connection to be Ricci semisymmetric, based on the choice of curvature tensors. Characterizations are then derived, and classifications are made under certain conditions.

Joint Projection Learning and Tensor Decomposition-Based Incomplete Multiview Clustering.

Incomplete multiview clustering (IMVC) has received increasing attention since it is often that some views of samples are incomplete in reality. Most existing methods learn similarity subgraphs from original incomplete multiview data and seek complete graphs by exploring the incomplete subgraphs of each view for spectral clustering. However, the graphs constructed on the original high-dimensional data may be suboptimal due to feature redundancy and noise. Besides, previous methods generally ignored the graph noise caused by the interclass and intraclass structure variation during the transformation of incomplete graphs and complete graphs. To address these problems, we propose a novel joint projection learning and tensor decomposition (JPLTD)-based method for IMVC. Specifically, to alleviate the influence of redundant features and noise in high-dimensional data, JPLTD introduces an orthogonal projection matrix to project the high-dimensional features into a lower-dimensional space for compact feature learning. Meanwhile, based on the lower-dimensional space, the similarity graphs corresponding to instances of different views are learned, and JPLTD stacks these graphs into a third-order low-rank tensor to explore the high-order correlations across different views. We further consider the graph noise of projected data caused by missing samples and use a tensor-decomposition-based graph filter for robust clustering. JPLTD decomposes the original tensor into an intrinsic tensor and a sparse tensor. The intrinsic tensor models the true data similarities. An effective optimization algorithm is adopted to solve the JPLTD model. Comprehensive experiments on several benchmark datasets demonstrate that JPLTD outperforms the state-of-the-art methods. The code of JPLTD is available at https://github.com/weilvNJU/JPLTD.

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Certain results on tangent bundle endowed with generalized Tanaka Webster connection (GTWC) on Kenmotsu manifolds

<p>This work studies the complete lifts of Kenmotsu manifolds associated with the generalized Tanaka-Webster connection (GTWC) in the tangent bundle. Using the GTWC, this study explores the complete lifts of various curvature tensors and geometric structures from Kenmotsu manifolds to their tangent bundles. Specifically, it examines the complete lifts of Ricci semi-symmetry, the projective curvature tensor, $ \Phi $-projectively semi-symmetric structures, the conharmonic curvature tensor, the concircular curvature tensor, and the Weyl conformal curvature tensor. Additionally, the research delves into the complete lifts of Ricci solitons on Kenmotsu manifolds with the GTWC within the tangent bundle framework, providing new insights into their geometric properties and symmetries in the lifted space. The data on the complete lifts of the Ricci soliton in Kenmotsu manifolds associated with the GTWC in the tangent bundle are also investigated. An example of the complete lifts of a $ 5 $-dimensional Kenmotsu manifold is also included.</p>