The paper studies automorphisms of RAAGs and RACGs, proving properties of their fixed subgroups.
problem Fixed subgroups of automorphisms of RAAGs and RACGs.
method Introducing coarse-median preserving automorphisms and proving properties of fixed subgroups.
result Fixed subgroups of RAAGs and RACGs are finitely generated, undistorted, and quasi-convex.
Study of automorphisms and splittings of special groups, showing infinite groups under certain conditions.
problem Understanding the structure and automorphisms of special groups G. method Constructing and analyzing non-small, stable G-actions on R-trees. result Conditions for the existence of infinite-order automorphisms and splittings.
Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
Generators found for automorphisms of special groups.
problem Characterizing automorphisms of special groups.
method Using a novel hierarchical shortening argument, the paper provides generators for automorphisms of special groups.
result Generators for automorphisms of special groups are found, including Dehn twists and pseudo-twists.
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
Study growth rates of automorphisms of special groups.
problem Understanding the growth rates of automorphisms of special groups.
method Analyzing outer automorphisms of virtually special groups, showing polynomial or exponential growth, and constructing Nielsen-Thurston decompositions.
result Outer automorphism groups of virtually special groups are boundary amenable, have finite virtual cohomological dimension, and satisfy the Tits alternative.
We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have the coarse median property defined by Bowditch. Consequences for such graphs include a distance formula analogous to Mas…
Study on connectivity and geometry of random Coxeter groups.
problem Connectivity threshold for square percolation on random graphs.
method Probabilistic combinatorics and techniques from geometric group theory.
result Determines connectivity threshold and cubical coarse median structure for random Coxeter groups.
Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting pro…
Algorithm confirms non-order-preserving braids, proving infinite family not order-preserving.
problem Determining if a braid is non-order-preserving.
method Algorithm that checks non-order-preserving property of braids.
result Infinite family of simple 3-braids are not order-preserving.
The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.
problem Investigating the evolution of planar curves with singularities under area-preserving and length-preserving inverse curvature flow.
method Area-preserving and length-preserving inverse curvature flow for ℓ-convex Legendre curves. result The flow results in a circle for ℓ-convex Legendre curves, providing geometric inequalities. Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
Classifies geodesic-preserving bijections in Thurston geometries.
problem Identifying bijections that preserve geodesics in different geometries.
method Comprehensive classification and proof for various geometries.
result Complete classification of geodesic-preserving bijections in Thurston geometries.
The paper studies quasimorphisms on density-preserving diffeomorphisms of the Möbius band.
problem Exploring quasimorphisms on groups of diffeomorphisms of non-orientable manifolds.
method Investigates the group of density-preserving diffeomorphisms on the Möbius band and shows the existence of unbounded quasimorphisms.
result The group of density-preserving diffeomorphisms on the Möbius band admits countably many unbounded quasimorphisms.
The paper examines flows that preserve area and length in hyperbolic geometry.
problem Preserving area and length in hyperbolic geometry.
method Inverse curvature flows for convex curves in hyperbolic plane.
result The flows converge to geodesic circles under certain conditions.
We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump-diffusion models. This …
Isometries of metric spaces (X,d) preserve all level sets of d. We formulate and prove cases of a conjecture asserting if X is a complete Riemannian manifold, then a function f:X→X preserving at least one level set d−1(r), with r>0 small enough, is an isometry.
This work preserves linear invariants in ensemble filters for non-Gaussian data assimilation.
problem Maintaining critical invariants like mass, stoichiometric balance, and charge in non-Gaussian data assimilation.
method Introducing a novel class of nonlinear ensemble filters using measure transport theory.
result Recovery of a constrained Kalman filter for Gaussian settings and combination with regularization techniques.
Volume-preserving neural networks prevent gradient issues.
problem Vanishing and exploding gradients in deep neural networks.
method A new neural network architecture with volume-preserving sublayers.
result Volume-preserving neural networks maintain gradient stability.
BiLipschitz mappings can be extended to preserve area.
problem Extending biLipschitz mappings to preserve area.
method Proving biLipschitz mappings can be extended to biLipschitz mappings preserving area.
result BiLipschitz mappings can be extended to preserve area.
For any α>0, we study kα-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in C∞-norm. Other relevant kα-type nonlocal flow is also discussed when α≥1.
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Turing complete flow on 4-sphere preserves volume.
problem Creating a Turing complete flow on a 4-sphere.
method Smooth, conservative flow on the 4-sphere.
result Achieved a Turing complete, volume-preserving flow.
Study proves stability and uniqueness for a specific type of flow.
problem Volume-preserving mean curvature flow stability and uniqueness.
method New gradient flow calibrations for volume preservation, stability estimate in distributional solutions.
result Strong solutions are calibrated and stable under certain conditions.
Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-k projection can be preserved using a smaller sketch of the original data matrix. We present a general structural result outlining four sufficient conditions to achieve projection-cost preservation. These condit…
In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermit…
Domain adaptation aims to assist the modeling tasks of the target domain with knowledge of the source domain. The two domains often lie in different feature spaces due to diverse data collection methods, which leads to the more challenging task of heterogeneous domain adaptation (HDA). A core issue of HDA is how to pre…
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
Sharp 3D Alexandrov inequality applied to volume-preserving flows.
problem Volume-preserving geometric flows in 3D space.
method Sharp quantitative Alexandrov inequality for C2-regular sets. result Established a 3D sharp quantitative version of the Alexandrov inequality.
New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
Preserving the privacy of individuals by protecting their sensitive attributes is an important consideration during microdata release. However, it is equally important to preserve the quality or utility of the data for at least some targeted workloads. We propose a novel framework for privacy preservation based on the …
The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.
problem Extending the measure preserving property to Moran sets.
method Analyzing bi-Lipschitz maps between Moran sets.
result Bi-Lipschitz maps between Moran sets preserve measure locally.
Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.
problem What conditions guarantee global existence of Gage's area-preserving flow for nonconvex initial curves?
method Using Dittberner's singularity analysis theory, constructed a ``flying wing'' curve to show limitations.
result Gage's area-preserving flow does not always preserve star-shapedness of evolving curves.
Hausdorff reflection keeps space shape intact.
problem Preserving shape type in spaces.
method Hausdorff reflection method.
result Hausdorff reflection preserves shape type.
We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is C3−close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such an end is foliated by spheres of Willmore type. In this paper, we prove that the leaves of this foliation are stable under sm…
Extends Arnold's linking theory to higher dimensions and submanifolds.
problem Volume-preserving actions in higher dimensions and submanifolds.
method Generalization of V. Arnold's theory to Rk and Rℓ. result Extension of asymptotic linking to higher dimensions and submanifolds.
Paper shows L∞-positivity and stochastic completeness are equivalent.
problem Analyzing L∞-positivity preserving property and stochastic completeness. method Using monotone approximation results for distributional solutions of −Δ+1≥0. result The L∞-positivity preserving property is equivalent to stochastic completeness. Study of parabolic-preserving deformations of hyperbolic lattices.
problem Preserving parabolic subgroups during lattice deformations.
method Analysis of deformations into SU(n,1) and SO(n+1,1).
result Existence of 1-parameter families of parabolic-preserving deformations.
The paper characterizes measures preserving independence through planar web geometry.
problem Characterizing measures with preserved independence.
method Planar web geometry and inhomogeneous Abelian functional equations.
result The independence-preserving property is preserved by coordinatewise reparametrizations and forms a natural invariant.
New curve flow preserves area and converges to a circle.
problem Preserving area while evolving curves to a circle.
method Area-preserving curvature flow for convex planar curves.
result The curve converges to a circle in smooth sense over time.
In this paper, we study global existence and blow up properties to Lp norm preserving non-local heat flows. We first study two kinds of Lp norm preserving non-local flows and prove that these flows have the global solutions. Finally, we give a example to show that one kind of this heat flow may blow up in $L^{\in…
In this paper, we introduce a flow over the projective bundle p:P(E∗)→M, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle OP(E∗)(1) is preserved along this flow under the null eige…
Reproduces IVFS for high-dimensional data structure preservation.
problem Preserving high-dimensional data structure in unsupervised feature selection.
method Inspired by random subset method, IVFS maintains data similarity through topological structure.
result IVFS outperforms SPEC and MCFS on most datasets.
Study a flow preserving area of plane curves, ending in a circle.
problem Preserving area while deforming plane curves.
method Non-local flow of convex closed plane curves.
result Limiting curve is a circle in the C∞ metric. Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.
problem Understanding the group of volume preserving diffeomorphisms through symplectic geometry.
method Using cotangent bundles of spaces of smooth embeddings, symplectic reduction, and nonlinear Grassmannians of augmented submanifolds.
result Descriptions of coadjoint orbits of the group of volume preserving diffeomorphisms in terms of submanifolds of augmented spaces.