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.
A space curve in a Euclidean 3-space E3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…
Study characterizes k-rectifiable sets in homogeneous groups.
problem Characterizing k-rectifiable sets in arbitrary homogeneous groups. method Proves characterizations using (k,G)-approximate tangent groups. result Existence of (k,G)-approximate tangent groups implies k-rectifiability. 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 F-connected complexes. 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…
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 (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. 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. 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.
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 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 this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a …
In Heisenberg groups, rectifiability is studied for subsets using C1,α-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 C1,α-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.
The paper proves smoothness of stationary varifolds.
problem Understanding the smoothness of stationary varifolds.
method Analyzing m-dimensional integer rectifiable varifolds in open sets. result The support of stationary varifolds is C∞ rectifiable. In this paper, we introduce a new class of curves αcalled a f-rectifying curves, which its f-position vector defined by α_{f}(s)=\int f(s)T(s)ds always lie in the rectifying plane of α, where f is an integrable function and T is the speed curve of α. In particular case, when the function f=0 or constant, the class of f…
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 ildeO(ε−2), matching optimal rate for mean estimation. 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.
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.
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.
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 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.
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.
In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We characterize rectifying curves in the n-dimensional Euclidean space in different ways…
Paper proposes a new activation function to reduce overfitting and large weight update issues.
problem Overfitting and large weight update problems in neural networks.
method Introduces a new activation function called Thresholded Exponential Rectified Linear Units (TERELU).
result TERELU shows better performance in reducing overfitting and large weight update issues compared to other activation functions.
Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.
problem Investigating minimizers of Ginzburg-Landau functionals in high dimensions with energy bounds.
method Analyzing minimizers with logarithmic energy bounds and considering the vacuum manifold's homotopy classes.
result Normalized energy measures converge to an (n−2)-rectifiable measure associated with a stationary varifold. This paper studies rectifiability in Carnot groups and proves geometric area formulas.
problem The study of rectifiability in Carnot groups and related geometric properties.
method Analysis of rectifiable measures in Carnot groups, geometric area formulas, and rectifiability of geodesic spheres.
result Geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in Carnot groups.
Study on harmonic maps in special geometric spaces.
problem Harmonic maps from rectifiable spaces into $\CAT(1)$ balls.
method Proving the existence and uniqueness of minimizers for energy function.
result Existence and uniqueness of minimizers for Korevaar-Schoen energy.
In this paper, we define a rectifying spacelike curve in the Minkowski space-time E14 as a curve whose position vector always lies in orthogonal complement N⊥ of its principal normal vector field N. In particular, we study the rectifying spacelike curves in E14 and characterize such curves in terms of…
The paper generalizes a theorem about rectifiability of sets.
problem Understanding the rectifiability of sets in geometric analysis.
method Generalizing a classical theorem of Besicovitch to new contexts.
result Sets with certain properties are rectifiable.
Let (M,g,f) be a 3-dimensional complete steady gradient Ricci soliton. Assume that M is rectifiable, that is, the potential function can be written as f=f(r), where r is a distance function. Then, we prove that M is isometric to (1) a quotient of R3, or (2) the Bryant soliton. In particular, we sh…
Rectangular peg problem solved for many curves.
problem Rectangular peg problem for continuous Jordan curves.
method Microlocal sheaf theory and recent work of Greene and Lobb.
result Affirmative answer for a large class of rectifiable curves.
CRITS improves time series classification with interpretable local explanations.
problem Lack of detailed explanations in time series classification models.
method CRITS uses convolutional kernels, max-pooling, and rectified linear units to extract feature weights.
result CRITS provides intrinsically interpretable local explanations without requiring gradients or random perturbations.
We show that any d-Ahlfors regular subset of Rn supporting a weak (1,d)-Poincaré inequality with respect to surface measure is uniformly rectifiable.
We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow.
We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (n…
Rectifies flat singular points for area-minimizing currents.
problem Understanding singularities of area-minimizing currents.
method Analyzes countably (m−2)-rectifiable singular points with flat tangent cones. result The set of singular density-Q points is countably (m−2)-rectifiable and has finite upper Minkowski content. We defined normal and rectifying curves in Pseudo-Galilean Space G_3^1. Also we obtained some characterizations of this curves in G_3^1.
Rectifies flat singular points of area-minimizing currents with singularity degree > 1.
problem Rectifying flat singular points of area-minimizing currents with singularity degree > 1.
method Subdividing singular points based on singularity degree and proving rectifiability of points with singularity degree > 1.
result The set of points with singularity degree > 1 is (m-2)-rectifiable.
A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve…
Paper proposes efficient image inversion and editing using rectified stochastic differential equations.
problem Inversion and editing of real images using generative models.
method Proposes RF inversion using dynamic optimal control and a linear quadratic regulator, extending to stochastic sampler for Flux.
result Allows state-of-the-art performance in zero-shot inversion and editing, outperforming prior works.
In this article, we show that, for any compact 3-manifold, there is a C1 volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the exam…
Proves rectifiability for specific metric spaces with unique tangents.
problem Rectifiability of CD(K,N) and MCP(K,N) spaces with unique tangents. method Failure of CD condition in sub-Finsler Carnot groups, new result on MCP spaces, recent breakthrough by Bate. result Proves rectifiability for CD(K,N) and MCP(K,N) spaces under specific conditions. In this paper, we have first given easily the characterization of special curves with the help of the Rotation minimizing frame (RMF). Also, rectifying-type curves are generalized n-dimensional space Rn.
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.