Study natural operators transforming tensor fields, proving all bilinear ones are of order one.
problem Understanding natural differential operators on tensor fields.
method Proved all bilinear operators are of order one, then classified operators in specific cases.
result Full classification of natural differential operators on tensor fields.
A Matlab toolbox for tensor operations based on t-product.
problem Extending matrix operations to tensors.
method Developed a Matlab toolbox implementing tensor operations based on t-product.
result Implemented several tensor operations including SVD, spectral norm, and nuclear norm.
Study invariant operations on Fedosov manifolds.
problem None explicitly stated in the abstract.
method Analysis of tensor-valued operations on Fedosov manifolds.
result Spaces of homogeneous natural tensors are finite dimensional linear representations of the symplectic group.
OpEvo automates tensor operator optimization for better efficiency.
problem Manual optimization of tensor operators is inefficient and limited.
method OpEvo uses evolutionary computation with topology-aware mutation.
result OpEvo finds optimal configurations with less effort and variance.
Completes the proof of curvature tensor existence for Jacobi operators.
problem Existence of curvature tensor for given Jacobi operators.
method Complete and accurate proof of the theorem, including a generalization to indefinite scalar product spaces.
result A complete proof of the existence of curvature tensor for given Jacobi operators, with a generalization.
Study of hypersurfaces in curved spaces with specific curvature properties.
problem Characterizing hypersurfaces in spaces of constant curvature with particular curvature properties.
method Investigates hypersurfaces isometrically immersed in semi-Riemannian spaces of constant curvature, focusing on the curvature tensor and its properties.
result Hypersurfaces in the specified spaces satisfy a Roter type equation, linking their curvature tensor to specific tensor products.
A new sketching method reduces tensor memory usage and enables efficient tensor operations.
problem Efficiently compressing and retaining tensor structure in large datasets.
method Higher-order Count Sketch (HCS) using multiple hash functions and tensor products.
result HCS achieves significant memory savings and efficient tensor operations.
The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.
problem Understanding Stein-Weiss operators on symmetric tensors of arbitrary rank.
method Analyzing the decomposition of tensor spaces into irreducible components and computing Weitzenbock formulas.
result Unified framework for second-order Stein-Weiss operators and tools for geometric analysis.
Study spectral functionals on manifolds with torsion.
problem Extending classical results to geometries with torsion.
method Using Wodzicki residue, investigate spectral functionals of Dirac and Laplace-type operators.
result Local densities recover fundamental geometric tensors.
We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.
Faster neural networks with approximate tensor operations reduce computation and training time.
problem Efficiently training deep neural networks with reduced computational and communication costs.
method Sample-based approximation of tensor operations (matrix multiplications and convolutions) to reduce computation and communication.
result Up to 66% reduction in computations and up to 1.37x faster training time with negligible accuracy loss.
Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
Study well-posedness of Faraday tensor problem on specific spacetime manifolds.
problem Well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary.
method Existence of Green operators for the operator d+δ and a suitable pre-symplectic structure on the space of solutions. result Existence of Green operators and pre-symplectic structure for the operator d+δ. The paper classifies tensors on specific Lorentzian metrics.
problem Classification of tensors on homogeneous plane waves.
method Framework of BGG operators to derive explicit formulae.
result Explicit formulae for irreducible Killing and conformal Killing 2-tensors identified.
We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence …
Proves elliptic operator images are closed on Hilbert bundles.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.
Framework learns to optimize tensor programs for various hardware.
problem Manual optimization of tensor operators for deep learning limits applicability and increases engineering costs.
method Learning-based statistical cost models guide tensor operator implementations over billions of variants.
result Framework delivers performance competitive with hand-tuned libraries across multiple hardware targets.
The paper extends comparison theorems using a new tensor field and operator.
problem Extending comparison theorems in Riemannian geometry.
method Using a divergence type operator and an extended Ricci tensor.
result Upper bounds for the end of manifolds and new theorems.
An odd vector field Q on a supermanifold M is called homological, if Q2=0. The operator of Lie derivative LQ makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called character…
Formula extended for elliptic operators, yielding eigenvalue estimates.
problem Estimating eigenvalues of elliptic differential operators.
method Proved Reilly formula for a specific class of elliptic operators.
result Derived estimates for the first positive eigenvalue.
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.
New formula extracts full local information from ray transform data.
problem Determining symmetric tensor fields from ray transform data.
method Deriving explicit formula for Saint Venant operator.
result Explicit formula for extracting full local information.
Using Weitzenböck techniques on any compact Riemannian spin manifold we derive a general inequality depending on a real parameter and joining the spectrum of the Dirac operator with terms depending on the Ricci tensor and its first covariant derivatives. The discussion of this inequality yields vanishing theorems for t…
Generalizes quantum integrability to all signatures for projectively equivalent metrics.
problem Quantum integrability for Beltrami-Laplace operators across various signatures.
method Shows that Killing tensors constructed from projectively equivalent metrics correspond to commuting differential operators.
result Quantum integrability for Beltrami-Laplace operators is established for all signatures.
Inverts rank m symmetric tensor fields using line integrals.
problem Recovering symmetric tensor fields from line integrals.
method Computes normal operator and presents inversion formula.
result Recovering rank m tensor fields from data (Nm0f,…,Nmmf). Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. This survey article studies when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of definition. It deals with results for the Jacobi operator, the higher order Jacobi opera…
Study on real hypersurfaces in complex quadric space with commuting structure Jacobi operator.
problem Characterizing real hypersurfaces in Qm with commuting structure Jacobi operator. method Analyzing structure Jacobi operator and Reeb curvature properties.
result Tube around CPk⊂Qm is the only Reeb flat Hopf hypersurface with commuting Ricci tensor and shape operator. Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If π is a spacelike 2 plane, let R(π) be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hyp…
We relate canonical algebraic curvature tensors that are built from a self-adjoint (RAS) or skew adjoint (RAΛ) linear operator A. Several authors have proven that any algebraic curvature tensor R may be expressed as a sum of RAS, or as a sum of RAΛ. This motivates our interest in relating them as well…
We study the asymptotic of the spectrum of the \spin Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin-Uribe, which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin.
We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also stu…
We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacob…
Computes indices of mixed order Dirac-type operators and related tensor fields.
problem Computing indices of mixed order Dirac-type operators and tensor fields.
method Using Hilbert complexes and differential operators of mixed order, computing indices with cohomology groups of tensor fields.
result Computation of indices for elasticity and biharmonic complexes.
We prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2].
Defines vector Laplacian on statistical manifolds.
problem No specific problem stated; focuses on mathematical definition.
method Defines and derives vector Laplacian formula.
result Derives formula for vector Laplacian.
Researchers describe local properties of Haantjes operators.
problem Understanding Haantjes operators with vanishing torsion.
method Complete local description of gl-regular Haantjes operators.
result Complete local description of gl-regular Haantjes operators.
Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields on M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…
Local fractional derivatives affect Riemann curvature tensor to zero.
problem Investigating how local fractional derivatives influence the Riemann curvature tensor.
method Introduced a general local fractional derivative operator and defined a specific Riemannian metric tensor field.
result The Riemann curvature tensor of the new metric is identically zero, indicating local isometry to Euclidean space.
The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally …
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric…
In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical me…
New characterization of Osserman tensors using Jacobi-orthogonality.
problem Characterizing Osserman tensors.
method Introducing Jacobi-orthogonality as a new potential characterization.
result Jacobi-orthogonal tensors are Osserman, and all known Osserman tensors are Jacobi-orthogonal.
New characterizations of ruled real hypersurfaces in complex projective space found.
problem Characterizing ruled real hypersurfaces in complex projective space.
method Defined tensor fields related to Levi-Civita and generalized Tanaka-Webster connections and studied the structure operator.
result Obtained new characterizations of ruled real hypersurfaces in complex projective space.
Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
Researchers factorize Dirac operators on toric noncommutative manifolds, finding curvature terms.
problem Factorizing Dirac operators on toric noncommutative manifolds.
method Using unbounded KK-theory and Kasparov modules, they show tensor sums of operators coincide with the Dirac operator on the manifold.
result There is a curvature term that arises as an obstruction for tensor sum decomposition in unbounded KK-theory.
Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
Study geodesic ray transform on 2D manifolds with conjugate points.
problem Understanding geodesic ray transform on manifolds with conjugate points.
method Decomposition into pseudodifferential operator and Fourier integral operators using method of stationary phase.
result Explicit computation of principal symbol and cancellation of singularities.
New eigenvalue estimate for CR manifolds' Kohn-Dirac operator.
problem Estimating eigenvalues of the Kohn-Dirac operator on CR manifolds.
method Characterizing equality case by CR twistor spinor existence; classifying manifolds with specific Ricci tensor properties.
result Classifying CR manifolds with at most two Webster Ricci tensor eigenvalues.