Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
Polluting fine dusts in South Korea which are mainly consisted of biomass burning and fugitive dust blown from dust belt is significant problem these days. Predicting concentrations of fine dust particles in Seoul is challenging because they are product of complicate chemical reactions among gaseous pollutants and also…
Match van Stockum dust to vacuum metrics with a single parameter.
problem Matching van Stockum dust to vacuum metrics.
method 1-parametric family of non-static Papapetrou vacuum metrics, Ehlers and Kramer--Neugebauer transformations.
result Explicit examples of matching, including Bonnor metric and Lanczos--van Stockum dust metric.
The study proves strong cosmic censorship violation for spherically symmetric dust clouds.
problem Violation of strong cosmic censorship for spherically symmetric dust clouds.
method Derived an ordinary differential equation for light rays and used it to prove strong cosmic censorship violation.
result Generic violation of strong cosmic censorship for spherically symmetric dust clouds.
3D dust map of the Milky Way improves resolution and accuracy.
problem Reconstructing the 3D dust distribution in the Milky Way.
method Gaussian process regression on spherical coordinates with iterative grid refinement.
result Improved 3D dust map with increased resolution and accuracy.
Study of caustics in Einstein-dust system, showing spacetime singularities and diverging curvature.
problem Understanding caustics and singularities in the Einstein-dust system.
method Established local existence result for spherically symmetric spacetimes containing caustics, constructed from solutions to a PDE problem.
result Obtained spherically symmetric spacetimes with diverging curvature and singular boundary.
Study of convergence of point-object configurations to a charged dust continuum.
problem Understanding the convergence of discretized point-object configurations to a charged dust continuum.
method Establishing existence and uniqueness of horizons/minimal surfaces, studying geometries of regions exterior to minimal surfaces, and discussing limits.
result Examples of scalar curvature jumps upon taking Gromov-Hausdorff and intrinsic flat limits.
New proof shows abelian Cantor groups can act on spaces.
problem Understanding actions of Cantor groups on metric spaces.
method Examined actions of abelian Cantor groups on metric spaces.
result Cantor groups can be abelian for n>1 in space actions.
Study timelike bounce in charged null dust collapse, identifying key surfaces.
problem Understanding charged null dust collapse dynamics and bounce surfaces.
method Novel decoupling of equations, constructing spacetime models, solving free boundary problems.
result Timelike bounce surfaces identified in charged null dust collapse, including examples terminating in null points.
A non-elementary Möbius group generated by two-parabolics is determined up to conjugation by one complex parameter and the parameter space has been extensively studied. In this paper, we use the results of \cite{GW} to obtain an additional structure for the parameter space, which we term the {\sl two-parabolic space}. …
We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeo…
We show that there are isometrically nonequivalent Robertson-Walker metrics which have the same set of geodesics. While one of these metrics satisfies the Einstein equations of pure dust without a cosmological constant, all the other describe pure dust with additional energy momentum tensor of cosmological constant typ…
We found a new simple family of Cantor sets whose projections are one-dimensional.
problem Finding simple Cantor sets with specific projection properties.
method Developed a new series of self-similar Cantor sets in R3. result All projections of these new Cantor sets are connected and one-dimensional.
In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the cont…
Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger…
First constructed genus 2 Cantor set in 3D space.
problem Constructing a geometrically self-similar Cantor set of genus 2.
method Geometrically self-similar construction in R3. result First uniformly quasiregular mapping with a genus 2 Cantor set Julia set.
By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension n can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension n. Moreover, the action can be assumed to be free if $n=…
The group of C1-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations nV of Thompson's group V arise…
All projections of typical Cantor sets in high dimensions are Cantor sets.
problem Whether all projections of a typical Cantor set in high dimensions are Cantor sets.
method Proving that for a dense Gδ subset of Cantor sets, all projections into non-zero linear subspaces are Cantor sets.
result There exists a dense Gδ subset of Cantor sets such that all projections into non-zero linear subspaces are Cantor sets.
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
Non-ergodic geodesic flow on Cantor tree surfaces found.
problem Determining when geodesic flow on Cantor tree surfaces is non-ergodic.
method Interpolating between two rates of convergence of cuff lengths to zero to prove non-ergodicity.
result Cantor tree surfaces with certain rates of cuff length convergence are non-parabolic.
New Cantor sets with high-dimensional projections discovered.
problem Understanding projections of Cantor sets in high dimensions.
method Construction and analysis of Cantor sets in Rn. result Cantor sets can be moved to have (n−2)-dimensional projections in (n−1)-planes. In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting…
Hyperbolic 3-manifolds can be approximated by removing Cantor sets from the 3-sphere.
problem Approximating hyperbolic 3-manifolds using Cantor set complements in the 3-sphere.
method Using exhaustion by π1-injective sub-manifolds and removing Cantor sets. result Hyperbolic 3-manifolds can be geometrically approximated by removing Cantor sets from the 3-sphere.
A classical theorem of Alexandroff states that every n-dimensional compactum X contains an n-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds,…
We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric.
For each Cantor set C in R3, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in R3 with complement having the same fundamental group as the complement of C. This answers a question from Open Problems in Topology and has as an application a simple c…
The study confirms most Cantor sets are in general position for all projections.
problem Understanding the general position of Cantor sets under various projections.
method Proof of the theorem stated in the title.
result Most Cantor sets are in general position with respect to all projections.
New minimal surfaces found with Cantor ends in convex domains.
problem Finding complex structures for minimal surfaces with Cantor ends.
method Proving existence of complete minimal surfaces with Cantor ends in minimally convex domains.
result Existence of a Cantor set whose complement forms a complete minimal surface.
We construct a large class of pathological n-dimensional topological spheres in Rn+1 by showing that for any Cantor set C⊂Rn+1 there is a topological embedding f:Sn→Rn+1 of the Sobolev class W1,n whose image contains the Cantor set C.
We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements …
The paper finds conditions for pseudosymmetric spacetimes to be perfect fluids.
problem Characterizing pseudosymmetric spacetimes as perfect fluids.
method Analyzes generalized Robertson-Walker spacetimes, conformally flat spacetimes, and dust fluids.
result Conditions for pseudosymmetric spacetimes to be perfect fluids are established.
For every finitely generated abelian group G, we construct an irreducible open 3-manifold MG whose end set is homeomorphic to a Cantor set and with end homogeneity group of MG isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-m…
Uncountably many fibrations found on genus 2 handlebody.
problem Finding fibrations on specific 3-manifolds.
method Constructing fibrations with Cantor tree fibers.
result Uncountably many fibrations with non-conjugate monodromies.
The paper studies the moduli space of generalized Cantor sets and their properties.
problem Understanding the moduli space of generalized Cantor sets and their equivalence.
method Constructing generalized Cantor sets and studying their moduli space properties.
result There are uncountably many moduli spaces and most have vanishing volume.
CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.
problem Finding CMC-1 surfaces on compact Riemann surfaces.
method Uniform approximation theorems for holomorphic null curves in C2imesC∗. result Cantor set removal allows for CMCext−1 immersions in hyperbolic and de Sitter spaces. New criteria for Cantor set tameness and wildness via projections.
problem Characterize dimensions of projections of Cantor sets.
method Geometric measure theory and Baire category theory.
result New criteria for Cantor set tameness and wildness.
Study shows similar result to Margulis for Cantor set homeomorphisms.
problem Understanding groups of homeomorphisms of Cantor sets.
method Analogous to Margulis's proof for linear groups.
result Groups of homeomorphisms either preserve a measure or contain a free subgroup.
Every normal subgroup of Cantor tree's mapping class group is geometric.
problem Characterizing normal subgroups of mapping class groups.
method Generalized curve graph study and adaptation of Brendle-Margalit strategy.
result All normal subgroups of Cantor tree's mapping class group are geometric.
A subset of Rd is called "sticky" if it cannot be isotoped off of itself by a small ambient isotopy. Sticky wild Cantor sets are constructed in Rd for each d≥4.
The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.
problem Understanding the structure of normal subgroups in mapping class groups of surfaces with specific subsets.
method Proves two structure theorems: purity and inertia, characterizing normal subgroups.
result Characterizes finite-type normal subgroups of mapping class groups of surfaces with Cantor subsets.
We study direct limits of embedded Cantor sets and embedded \sier curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called ω-Cantor spaces, and similarly, all limits of \sier curves give homeomorphic spaces, called to ω-\sier curves. W…
Proposes MM-DUST for efficient generalized lasso solution paths.
problem Efficiently solve generalized lasso problems in large-scale and non-linear models.
method Majorization-minimization dual stagewise algorithm incorporating quadratic majorizers and stagewise learning.
result Established the uniform convergence of approximated solution paths.
The paper constructs wild Cantor sets in high dimensions.
problem Embedding Cantor sets in high-dimensional spaces.
method Constructing embeddings of Cantor sets in \(\mathbb{R}^n\).
result Embeddings create pairwise wild Cantor sets that are ambiently incomparable.
We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension <1 are free. On the other hand we construct for any ε>0 examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension <1+ε.
New method uses Cantor embeddings and Wasserstein distances to analyze predictive states in time series data.
problem Analyzing predictive states in stochastic processes using time series data.
method Wasserstein distances for detecting predictive equivalences in symbolic data, using Cantor embeddings for finite-dimensional representation.
result Exploratory analysis of temporal structure in various processes reveals insights.
Minimal surfaces can be mapped to 3D with bounded images.
problem Mapping minimal surfaces to 3D with bounded images.
method Analyzes various types of minimal immersions into R3 and complex manifolds. result Every surface contains a Cantor set allowing bounded conformal minimal immersions.
The paper proves existence of solutions for Einstein-type elliptic systems on AE manifolds.
problem Analyzing semi-linear systems of partial differential equations motivated by the conformal formulation of Einstein constraint equations.
method Proving existence theorems under suitable conditions, including smallness assumptions on free parameters.
result Existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations.