Study strong max principle for vector maps on curved spaces.
problem Characterize vector valued harmonic maps with linear growth.
method Apply strong max principle to maps on manifolds with non-negative Ricci curvature.
result Equalities among supremum, asymptotic average, and heat evolution of map norms.
The paper classifies maps from vector bundles to Euclidean spaces.
problem Classifying homotopy classes of maps from vector bundles to Euclidean spaces.
method Using homotopy theory and vector bundles.
result Computed homotopy classes of proper maps and stability range.
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
We discuss sharp Sobolev inequalities for vector valued maps.
A map between manifolds which matches up families of complete vector fields is a fiber bundle mapping on each orbit of those vector fields.
The normal map of curves is analyzed as a vector field on a cylinder.
problem Understanding the geometric properties of normal maps and their vector field interpretation.
method Interpreting critical points geometrically, studying Poincaré index, projecting to sphere, and analyzing winding and rotation indices.
result Counting theorems regarding winding and rotation indices of curves and their evolutes are proven.
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
Harmonic and minimal great circle fibrations have special Gauss maps.
problem Characterizing Gauss maps of harmonic and minimal great circle fibrations.
method Analyzing the relationship between the Gauss map and the generating unit vector field.
result The Gauss map of a great circle fibration is harmonic (minimal) if and only if the generating unit vector field is harmonic (minimal).
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Study of Pascal algebra matrices and their jet bundle map for vector bundles.
problem Defining and studying Pascal algebra matrices and their map on jet bundles.
method Identifying Pascal algebra matrices, showing generator well defines Pascal map, using it for intrinsic contact definition.
result Intrinsic definition of point-wise contact between Hermitian vector bundles using unitary equivalence of Pascal maps.
The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.
problem Existence of harmonic and bi-harmonic maps into certain Riemannian manifolds.
method Analysis of manifolds with conformal vector fields or Ricci solitons.
result Nonexistence of harmonic and bi-harmonic maps in specified conditions.
Shows Euler-like vector fields come from specific embeddings.
problem Understanding Euler-like vector fields and their origins.
method Using tubular neighborhood embeddings and normal exponential maps of Riemannian metrics.
result Each Euler-like vector field originates from a specific embedding.
The paper characterizes vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
problem Characterizing vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
method Characterization theorem and critical point condition for interpolating sesqui-harmonic vector fields.
result Conditions for vector fields to be interpolating sesqui-harmonic maps on compact manifolds.
We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.
Symphonic maps on non-compact Riemannian manifolds are non-existent.
problem Existence of symphonic maps on compact or non-compact Riemannian manifolds
method Study of conformal vector fields and Ricci solitons
result No symphonic maps exist
The study finds conditions for a special vector field in singularities.
problem Existence of Milnor vector field for new singularities.
method Introducing sufficient conditions for the existence of the Milnor vector field.
result Conditions guarantee the existence of the Milnor vector field for new classes of singularities.
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and L∞-algebras. result Extension of conserved currents to spacetime vector fields.
Neural networks fail to bridge modalities as expected.
problem Understanding how well neural networks map between different modalities.
method Proposed a new similarity measure and conducted experiments on cross-modal benchmarks.
result Predicted vectors do not resemble the target vectors' neighborhood structure.
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
Introduces MFVDM for high-dimensional data analysis.
problem Non-linear dimensionality reduction of high-dimensional datasets.
method Combines multiple unitary irreducible representations for nonlinear embeddings.
result Achieves better nearest neighbor search and alignment estimation on noisy data.
Functorial compactification of vector spaces defined as manifolds with corners.
problem Compactification of vector spaces functorial under linear maps.
method Definition of manifolds with corners and b-maps, application of iterated blow-up theory.
result Criterion for lifted maps to be b-fibrations, identification of restrictions to boundary hypersurfaces.
Generators for the module of vector fields liftable over corank 1 stable complex analytic maps from an n-manifold to an (n+1)-manifold are found. This is applied to the classification of the singularities occuring in generic one-parameter families of maps between these spaces.
New formulas for geometric measures in vector spaces.
problem Local additive kinematic formulas for vector spaces.
method Introducing dual area measures and proving their convolution product.
result Local additive kinematic formulas in hermitian vector spaces.
Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.
problem Finding zeros of mappings from a manifold into a vector bundle.
method Local convergence using differentiability concepts, Banach space Riemannian distance, and affine covariant damping strategy.
result Illustrated application to generalized non-symmetric eigenvalue problems.
Note on subgaussian bounds for sign-quantized linear maps.
problem Understanding subgaussian behavior of sign-quantized linear maps.
method Developed a dimension-independent subgaussian concentration bound for Gaussian vectors under nonlinear mappings.
result Answered a question about sign-quantized linear maps using a new subgaussian bound.
This research simplifies Riemannian LBFGS for SPD manifolds.
problem Optimization on Riemannian manifolds, especially SPD.
method Two mappings for tangent space, making vector transports and adjoint vector transports identity.
result RLBFGS becomes less computationally expensive and easier to analyze.
Complex functional maps link tangent bundles, preserving orientation and angles.
problem Linking tangent bundles for orientation-aware correspondence.
method Endow tangent bundles with complex structures to enable robust transfer of tangent vector fields.
result Establishes orientation-aware correspondence without relying on descriptors or extra regularization.
Classifies sections of Riemannian bundles on Lie groups.
problem Classifying sections of Riemannian bundles on Lie groups.
method Developed variational theory of higher-power energy for mappings and sections.
result Complete classification of left-invariant vector fields on 3D Lie groups.
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
LA-VDM accelerates VDM using landmarks to improve data analysis.
problem Efficiently analyzing complex datasets with nonuniform sampling densities.
method Landmark-constrained two-stage normalization to accelerate VDM.
result LA-VDM accurately recovers parallel transport and converges to the connection Laplacian.
Study on generalized ξ-parallel maps in Riemannian geometry.
problem Characterizing and understanding generalized ξ-parallel maps.
method Defined energy functional, derived first variation formula, and Euler-Lagrange equation.
result Established fundamental properties and relationships with harmonic and biharmonic maps.
This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori.…
The well-known AKSZ construction (for Alexandrov--Kontsevich--Schwarz--Zaboronsky) gives an odd symplectic structure on a space of maps together with a functional S that is automatically a solution for the classical master equation (S,S)=0. The input data required for the AKSZ construction consist of a volume eleme…
The article discusses conservation laws for polyharmonic maps and their applications.
problem Understanding conservation laws for polyharmonic maps.
method Recalling the stress-energy tensor and showing conservation laws with Killing vector fields.
result Conservation laws for polyharmonic maps and their applications.
A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a presc…
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…
We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details
The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…
Deep neural network improves speaker verification with short utterances.
problem Challenges in speaker recognition with short utterances.
method Proposes deep neural network mapping methods to improve i-vector performance.
result Deep neural network mapping significantly improves speaker verification performance with short utterances.
New unitary representations for mapping class groups without almost invariant vectors.
problem Understanding unitary representations of mapping class groups.
method Space of measured foliations and Teichmüller space.
result None of the representations has almost invariant vectors.
VDTW improves cross-year crop mapping accuracy.
problem Cross-year crop mapping accuracy is poor with existing methods.
method Vector Dynamic Time Warping (VDTW) for multi-year classification.
result VDTW achieves 99.85% and 99.74% overall accuracies for same and cross years, respectively.
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss map. We prove that a marginally trapped surface is of pointwise 1-type Gauss map if…
The study connects moment maps, star products, and automorphism groups on Kaehler manifolds.
problem Analyzing the structure of automorphism groups on Kaehler manifolds with specific curvature properties.
method Using star products, moment maps, and Hessian formulas to study holomorphic vector fields.
result Proves a reductive Lie algebra structure for holomorphic vector fields on Kaehler manifolds.
New statistical manifolds derived from identity map biharmonicity.
problem Deriving new statistical manifolds from identity map biharmonicity.
method Statistical biharmonicity of identity maps, semi-equiaffine condition, constant curvature.
result Determined statistical structures of new class of manifolds.
This paper simplifies deep learning networks by mapping them to a linear function of a feature map.
problem Understanding how weights in deep networks coordinate across layers and generalize.
method Reparameterizes DNNs as a linear function of a feature map, transforming depth-dependencies into tensor products.
result Develops sample compression representation of neural networks in terms of support vectors, showing sample complexity of O(ns/epsilon).
Study the structure of linear bundle morphisms between vector bundles.
problem Understanding the structure of linear bundle morphisms between vector bundles.
method Analyzing the set of smooth linear bundle morphisms between fibre bundles.
result Detailed structure of linear bundle morphisms between vector bundles.
Study topological invariants for hypersurfaces using vector fields.
problem Finding obstructions to foliations on hypersurfaces.
method Define a perturbed Gauss map to derive topological invariants.
result Obtained topological invariants that depend on manifold geometry.
VectorNet predicts car behavior using vectorized HD maps and agent dynamics.
problem Predicting behavior in multi-agent systems with self-driving cars.
method VectorNet uses hierarchical graph neural networks on vectorized representations of HD maps and agent trajectories.
result VectorNet achieves comparable or better performance than state-of-the-art methods while using fewer parameters and less computational power.