The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. Establishes a lower bound for Kähler-Einstein distance on certain domains.
problem Finding a lower bound for Kähler-Einstein distance on specific types of domains.
method Proves an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.
result Establishes a lower bound for the Kähler-Einstein distance on pseudoconvex domains with positive hyperconvexity index.
We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph Γ. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than 2π on time-like singular segments). We construct examples of such manifolds, d…
We introduce and study the flow of metrics on a foliated Riemannian manifold (M,g), whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, $\Sc_{\,\rm mix}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{\,\rm mix}$ (…
New Kazdan-Warner problem for equivariant metrics on manifolds.
problem Equivariant scalar curvature functions on manifolds with group actions.
method Established equivariant analogue of Kazdan-Warner trichotomy.
result New class of totally G-positive pairs with positive constant scalar curvature.
Paper develops heavy-tailed embeddings for better text classification and augmentation.
problem Improving text classification, especially for extreme values.
method Develops heavy-tailed embeddings using multivariate extreme value theory and introduces a scale-invariant classifier.
result The classifier outperforms baselines and generates meaningful augmented text.
We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph Γ. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than 2π on time-like singular segments). We construct examples of such manifolds, d…
Unsupervised representation learning has succeeded with excellent results in many applications. It is an especially powerful tool to learn a good representation of environments with partial or noisy observations. In partially observable domains it is important for the representation to encode a belief state, a sufficie…
Structured regularizers enable faster optimization on SPD manifolds with constraints.
problem Optimizing SPD matrices with additional constraints.
method Structured regularizers based on symmetric gauge functions.
result Structured regularizers can preserve or induce desirable structure like convexity.
Jointly estimates flow fields and particle properties from Lagrangian data.
problem Estimating flow fields and particle properties from sparse, noisy Lagrangian data.
method Data assimilation framework coupling Eulerian and Lagrangian models.
result Joint estimation of flow fields and particle properties in various flow regimes.
Speaker verification (SV) systems using deep neural network embeddings, so-called the x-vector systems, are becoming popular due to its good performance superior to the i-vector systems. The fusion of these systems provides improved performance benefiting both from the discriminatively trained x-vectors and generative …
Paper transforms torse-forming vector fields into simpler forms.
problem Generalizing vector fields and their transformations.
method Present techniques to transform torse-forming vector fields into simpler cases.
result Concrete examples of transformations are provided.
Given an d-dimensional manifold with two commuting Killing vectors, together with an d - 1 dimensional submanifold in which one of the Killing vectors lies, then the lapse and shift of the second Killing vector, relative to this slice, remain constant along the orbits of the `surface' Killing vector. Alternatively, the…
New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.
problem Proving triviality of modified conformal vector fields on Riemannian manifolds.
method Analyzing properties of homothetic, conformal, and gradient vector fields on compact and non-compact manifolds.
result Established conditions under which m-modified conformal vector fields are trivial. The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …
For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…
Characterizes spacetimes using doubly torqued vectors.
problem Classifying spacetimes based on their geometric properties.
method Characterization through doubly torqued vectors and their properties.
result Doubly twisted and Kundt spacetimes can be characterized.
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. Abstract: Generalizes multisymplectic forms to vector-valued versions.
problem Generalizing multisymplectic forms to vector-valued versions.
method Obtained a standard local presentation and proved an entropy inequality for partial compositions.
result Vector-valued multisymplectic forms form a non-unital operad.
The paper bounds the mean absolute error in DNN vector-to-vector regression.
problem Bounding the mean absolute error in deep neural network based vector-to-vector regression.
method Error decomposition techniques in statistical learning theory and non-convex optimization theory were used to derive upper bounds for approximation, estimation, and optimization errors.
result Theoretical upper bounds for mean absolute error in DNN vector-to-vector regression were derived and validated experimentally.
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. SVM generalizes well even with many support vectors in high dimensions.
problem Generalization of SVM in high-dimensional spaces with many support vectors.
method Identified new deterministic equivalences and proved conditions for support vector proliferation.
result Broadened conditions for SVM generalization in high-dimensional settings and proved converse result.
Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.
problem Optimal mass transport for vector-valued Gaussian mixtures.
method Vectorizing Gaussian mixture models and studying optimal mass transport problems.
result Computational efficiency and structure preservation in optimal mass transport.
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
We prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which, to any pair formed of a quaternionic vector space and a real subspace, associates a coherent sheaf over the sphere.
Study on vector fields on Lie groups reveals surprising algebraic coincidences.
problem Characterizing vector fields on Lie groups with Riemannian metrics.
method Algebraic and geometric analysis of left-invariant vector fields on nilpotent Lie groups.
result Spaces of Killing, one-harmonic, and conformal vector fields coincide with the center of the Lie algebra on nilpotent Lie groups.
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
This paper introduces ∞- and n-fold vector bundles as special functors from the ∞- and n-cube categories to the category of smooth manifolds. We study the cores and "n-pullbacks" of n-fold vector bundles and we prove that any n-fold vector bundle admits a non-canonical isomorphism to a decomposed …
Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Study on Einstein solitons with specific vector fields and their properties.
problem Characterizing Einstein solitons with gradient, solenoidal, or concircular vector fields.
method Explicitly express the function λ by gradient vector field V and deduce geometric properties under certain curvature conditions.
result Explicit expressions for λ and geometric properties of Einstein solitons.
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
In this paper we show that for an invariant (α,β)−metric F on a homogeneous Finsler manifold HG, induced by an invariant Riemannian metric a~ and an invariant vector field X~, the vector X=X~(H) is a geodesic vector of F if and only if it is a geodesic vector of a~. …
SL(n) covariant valuations on Orlicz spaces are represented and characterized.
problem Representing SL(n) covariant valuations on Orlicz spaces.
method Representation theorem established for continuous, SL(n) covariant vector-valued valuations.
result Unique characterization of SL(n) covariant valuations as moment vectors.
The paper explores MAE as a loss function for DNN vector-to-vector regression, proving its advantages over MSE.
problem Improving loss function for deep neural network based vector-to-vector regression.
method Presenting performance bounds and new properties of MAE, deriving generalized upper bounds, and interpreting MAE as a Laplacian distribution.
result MAE is a more suitable loss function than MSE for DNN based vector-to-vector regression, especially when errors follow a Laplacian distribution.
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
Investigates point spectra of vector fields and their properties.
problem Understanding the point spectra of vector fields.
method Define and study point spectra, prove properties under isometries, and analyze compactly supported fields.
result Point spectra are well-behaved under isometries and trivial for compactly supported fields.
We explore under what conditions one can obtain a nontrivial knot, given a collection of n vectors. First, we show how to get a crossing from any 3 vectors equal in magnitude, by arbitrarily picking 2 vectors and identifying the sufficient and necessary criteria for picking a third vector that will guarantee a crossi…
We consider four dimensional lie groups equipped with left invariant Lorentzian Einstein metrics, and determine the harmonicity properties of vector fields on these spaces. In some cases, all these vector fields are critical points for the energy functional restricted to vector fields. We also classify vector fields de…
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
The paper proves non-existence of torqued and anti-torqued vector fields on hyperbolic spaces.
problem Existence of torqued and anti-torqued vector fields on hyperbolic spaces.
method Analyzing the properties of conformal scalar functions and their impact on the existence of vector fields.
result Non-existence of proper torqued and anti-torqued vector fields on hyperbolic spaces.
We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are …
Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
Study classifies harmonic vector fields on 3-manifolds.
problem Classifying harmonic unit vector fields on 3-manifolds.
method Investigates under mild curvature assumptions, classifying vector fields and manifolds.
result Classifies both vector fields and manifolds supporting them.
Criteria for lifting manifold diffeomorphisms to vector bundle automorphisms.
problem Lifting diffeomorphisms to vector bundles.
method Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
result Criteria for lifting diffeomorphisms to linear automorphisms of vector bundles.
In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.
The support vector machine (SVM) is an important class of learning machines for function approach, pattern recognition, and time-serious prediction, etc. It maps samples into the feature space by so-called support vectors of selected samples, and then feature vectors are separated by maximum margin hyperplane. The pres…