We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…
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48 results for “rectifiable chains”
Study real rectifiable currents, generalize King's theorem, simplify proof, relate to Hodge conjecture.
problem Characterize currents defined by positive real holomorphic chains.
method Use Siu's semicontinuity theorem to simplify King's proof.
result Sufficient condition for the Hodge conjecture.
Study shows almost minimizing rectifiable chains in Hilbert space have regular points dense in their support.
problem Understanding the regularity of almost minimizing rectifiable chains in infinite dimensional spaces.
method Adapted Reifenberg's epiperimetric inequality and computations by Preiss to infinite dimensional space.
result The set of regular points is dense in the support of almost mass minimizing rectifiable chains.
The study provides bounds for geodesic diameter in Euclidean space.
problem Finding bounds for geodesic diameter in Euclidean space.
method Develops a geometric approach using locally rectifiable chains and complete normed commutative group bundles.
result Provides a new method for calculating geodesic diameter bounds.
Characterizes real holomorphic chains on complex manifolds.
problem Representing homology classes by algebraic cycles.
method Characterization of real holomorphic chains; application to homology classes.
result Real holomorphic chains are characterized by local properties.
We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…
In this paper we prove that, in the category of chain complexes, partial algebras can be functorially replaced by quasi-isomorphic algebras. In particular, partial algebras contain all of the important homological and homotopical information that genuine algebras do. Applying this result to McClure's partial algebra in…
The paper solves a partial Plateau problem using -mass.
problem Finding a surface of least area with a partially specified boundary.
method Minimizing -mass over scans with boundary.
result Existence of a rectifiable minimizer for the -mass problem.
This thesis is divided into three parts. In the first part, we give an introduction to J. Harrison's theory of differential chains. In the second part, we apply these tools to generalize the Cauchy theorems in complex analysis. Instead of requiring a piecewise smooth path over which to integrate, we can now do so over …
Develops a new method for solving sparse non-negative least squares problems.
problem Sparse non-negative least squares problem.
method Bayesian evidence maximization framework using Rectified Gaussian Scale Mixture (R-GSM) prior and Expectation-Maximization (EM) algorithm.
result Proposed R-SBL method outperforms existing solvers in signal and support recovery.
Two modifications improve classifier chains for multi-label classification.
problem Discrepancy between training and testing feature spaces in classifier chains.
method Proposed modifications to address attribute noise.
result Improved prediction performance in challenging cases.
Proposes a new Hodge conjecture in Bott-Chern cohomology.
problem Hodge conjecture in Bott-Chern cohomology.
method Characterization of real holomorphic chains, atomic section theory, refined Bott-Chern classes.
result Proof of a new Hodge conjecture in Bott-Chern cohomology.
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
Rectifying submanifolds are characterized by their tangential position vector field component.
problem Characterizing rectifying submanifolds in Euclidean spaces.
method Introducing rectifying submanifolds and proving their properties.
result Rectifying submanifolds are identified by a specific tangential vector field property.
A non-Newtonian approach in differential geometry of curves: multiplicative rectifying curvesmath.DG
Study rectifying curves in 3D multiplicative Euclidean space.
problem Investigate rectifying curves in a non-Newtonian geometry setting.
method Apply multiplicative differential-geometric concepts to rectifying curves.
result Classify multiplicative rectifying curves using spherical curves.
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
Study characterizes -rectifiable sets in homogeneous groups.
problem Characterizing -rectifiable sets in arbitrary homogeneous groups.
method Proves characterizations using -approximate tangent groups.
result Existence of -approximate tangent groups implies -rectifiability.
New curves generalize helix and rectifying curves.
problem Generalizing helix and rectifying curves.
method Introducing f-rectifying curves with f-position vector in rectifying plane.
result Classification and characterization of f-rectifying curves.
Recalls and refines the concept of algebraically rectifiable curves.
problem Classical notion of algebraically rectifiable plane curves.
method Provides new criteria, relates to quadratic differentials, and generalizes to higher order differentials.
result Generalization and new criteria for algebraic rectifiability.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into -connected complexes.
Uniform rectifiability proven for sets with Poincaré inequalities.
problem Uniform rectifiability of sets with Poincaré inequalities.
method Weak -Poincaré inequality and surface measure.
result Uniform rectifiability achieved for sets supporting such inequalities.
New rectified flow method improves image generation and converges to optimal transport.
problem Improving computational and statistical guarantees of rectified flow for image generation.
method Introducing c-rectified flow, which projects velocity fields onto a gradient class while preserving marginals.
result Iterative c-rectified flow always converges to the optimal transport coupling under suitable assumptions.
Paper transforms deep rectifier networks into shallow ones for analysis.
problem Understanding the complexity of deep neural networks.
method Transformation of deep rectifier networks into shallow ones.
result Shallow networks can represent deep networks with fewer functions.
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves -rectifiability of singular set.
result Singular set is -rectifiable.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying rectifying and normal curves in Lorentzian n-space.
method Introducing a position vector field and defining rectifying and normal curves based on this field.
result Comprehensive characterization and classification of rectifying and normal curves.
In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evalu…
Neural networks can approximate rectifiable measures with small error.
problem Approximating complex rectifiable measures using neural networks.
method Using ReLU neural networks to approximate (countably) m-rectifiable measures as push-forwards of the Lebesgue measure.
result The approximation error in terms of Wasserstein distance can be made arbitrarily small.
The study characterizes rectifying curves in n-dimensional space.
problem Understanding rectifying curves in arbitrary dimensions.
method Characterization through various conditions and constructions.
result Different ways to characterize rectifying curves in n-dimensional Euclidean space.
We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.
The paper characterizes special curves and generalizes rectifying-type curves in n-dimensional space.
problem Characterizing and generalizing special curves in higher dimensions.
method Characterization through Rotation minimizing frame (RMF) and generalization of rectifying-type curves.
result Rectifying-type curves are generalized in n-dimensional space.
Rectifies zero loci of certain spinors in 4D manifolds.
problem Understanding the structure of zero loci of harmonic spinors.
method Proves rectifiability and Minkowski bounds for the zero locus.
result Zero loci are 2-rectifiable and have locally finite Minkowski content.
The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and n_2 of the curve. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifyin…
In Heisenberg groups, rectifiability is studied for subsets using -regular surfaces.
problem Understanding rectifiability of subsets in Heisenberg groups.
method Introducing a new notion of rectifiability and proving conditions for rectifiability using tangent paraboloids.
result A sufficient condition for -rectifiability of low-codimensional subsets in Heisenberg groups is the existence of suitable approximate tangent paraboloids.
The paper characterizes rectifying curves on smooth surfaces using isometries and Darboux frames.
problem Characterizing rectifying curves on smooth surfaces under isometries.
method Using Darboux frames and isometries to investigate rectifying curves.
result Find deviations of rectifying curves under isometries and analyze their properties.
Rectifiable varifolds with bounded curvature can be covered by smooth surfaces.
problem Understanding the structure of rectifiable varifolds with bounded curvature.
method Using curvature of arbitrary closed sets and viscosity solutions of PDEs.
result The support of rectifiable varifolds can be covered by smooth submanifolds.
The paper proves smoothness of stationary varifolds.
problem Understanding the smoothness of stationary varifolds.
method Analyzing -dimensional integer rectifiable varifolds in open sets.
result The support of stationary varifolds is rectifiable.
New geometric definition for subsets of Euclidean space, proving second-order rectifiability.
problem Defining and proving second-order rectifiability for closed subsets of Euclidean space.
method Defining strata based on linear independence of touching directions and proving rectifiability using a new criterion.
result The -th stratum of a closed subset of an -dimensional Euclidean space is second-order rectifiable of dimension .
The paper explores rectified flows and their relation to optimal transport.
problem Understanding the connection between rectified flows and optimal transport.
method Investigates invariance properties, explicit constructions, and analysis of rectified flows in various settings.
result Rectified flows, when gradient constrained, do not generally solve optimal transport problems.
Rectified flows achieve optimal sample complexity for generating data.
problem Generating high-quality data samples efficiently.
method Rectified flows constrain transport trajectories to be linear, enabling efficient sampling.
result Achieve sample complexity of , matching optimal rate for mean estimation.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.
Improves AI-prior reliability for Bayesian inference.
problem Error propagation from predictive models into posterior inference.
method Rectified AI-informed prior elicitation framework.
result Significant reduction in bias and improvement in predictive performance.
The rectified flow method is analyzed for its statistical properties.
problem Theoretical support for rectified flow methods is lacking.
method Empirical analysis of rectified flow's statistical properties using regression and density estimation.
result Convergence rates for rectified flow estimators are faster than for nonparametric regression and density estimation.
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
Rectifying curves on hypercones are geodesics, characterized in higher dimensions.
problem Characterizing rectifying curves in higher-dimensional spaces.
method Extending results from Chen (2017) to higher dimensions, using hypercones and hyperplanes.
result Rectifying curves on hypercones are geodesics, and these curves can be mapped to spherical curves in higher dimensions.
Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitonsmath.DG
3D solitons classified into specific types.
problem Classifying 3D steady and expanding gradient Ricci solitons.
method Analyzing rectifiable potential functions and curvature conditions.
result 3D solitons are isometric to specific known forms.
Study rectifying submanifolds with anti-torqued axis in Riemannian manifolds.
problem Characterize submanifolds with anti-torqued axis in Riemannian manifolds.
method Determine necessary and sufficient conditions for anti-torqued vector fields, characterize submanifolds, and derive rectifying submanifolds as warped products.
result Rectifying submanifolds with anti-torqued axis are warped products with specific warping functions.