Smooth approximations near singularities of constant mean curvature surfaces are found.
problem Finding smooth approximations for constant mean curvature surfaces near singular points.
method Proving the existence of sequences of smooth CMC hypersurfaces converging to a given one in a ball centered at the singularity.
result Smooth approximations exist in a ball centered at the singularity of a CMC hypersurface.
Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. Paper shows a surface with positive genus under a tetrahedron, less than cone's area.
problem Existence of area-minimizing surfaces with positive genus spanning a tetrahedron.
method Triple cover approach, using BV functions and Caccioppoli sets.
result Found a surface of positive genus with area less than a conic surface.
The study extends removability results for quasiregular curves in Euclidean spaces.
problem Removability of singularities in quasiregular curves.
method Extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves.
result Every non-constant quasiregular curve has infinite energy.
This is a revised version (minor changes and a deeper insight in the positive curvature case). We prove some Caccioppoli's inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature …
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Cacciop…
Proves multiplicity one for min-max minimal hypersurfaces in specific manifolds.
problem Proving multiplicity one for min-max minimal hypersurfaces in specific manifolds.
method Using min-max theory for hypersurfaces with prescribed mean curvature and approximating min-max values.
result Confirms a conjecture by Marques-Neves for min-max minimal hypersurfaces in bumpy metrics.
New geometric interpretations reveal structure of AC integrands.
problem Understanding and verifying the atomic condition for integrands.
method Reinterpretation of atomic condition in convex geometry.
result Quantitative versions EC and QEC proposed; stability and regularity results.
Establishes interior regularity results for a broad class of two-dimensional nonlinear elliptic systems using a unified abstract framework.
problem Interior regularity results for two-dimensional nonlinear elliptic systems
method A unified abstract framework built around a Campanato-type discrete iteration scheme coupled with a Caccioppoli-type estimate
result Local Hölder continuity of the map u is established, with an explicit Hölder exponent that optimally attains the classical Morrey--Campanato threshold dictated by the Lebesgue integrability of the source term f Study on liquid-vapor interfaces in stable equilibrium without assuming prior regularity.
problem Understanding the conditions for a liquid-vapor interface to be in stable equilibrium without assuming prior regularity.
method Proposes a weakest set of mathematical assumptions using varifold regularity theory and identifies a suitable stability condition.
result The liquid-vapor interface is a smoothly embedded analytic surface in stable equilibrium.
The paper proves Hardy inequalities on Finsler manifolds using superharmonicity.
problem Establishing Hardy inequalities on Finsler manifolds.
method Using superharmonicity of a weight function and properties of the Finsler-Laplace operator.
result Generalization of Riemannian Hardy inequalities to Finsler manifolds.
The paper proves the existence of hypersurfaces with prescribed mean curvature.
problem Proving the existence of hypersurfaces with prescribed mean curvature.
method PDE theoretic approach using mountain pass construction and regularity results for integral varifolds.
result Existence of quasi-embedded, boundaryless hypersurfaces with prescribed mean curvature.
The paper extends CMC varifolds theory to prescribed mean curvature in Riemannian manifolds.
problem Developing a regularity and compactness theory for varifolds with prescribed mean curvature.
method Generalizing previous results to Riemannian manifolds, identifying conditions for regularity and compactness.
result The support of the varifold is the image of a C2 immersion with continuous unit normal and mean curvature g. Alternative proof and extension of curvature estimates for minimal immersions.
problem Curvature estimates and Bernstein-type theorems for minimal immersions.
method Iteration method à la De Giorgi, ε-regularity theorem, Caccioppoli inequalities.
result Extension of Schoen--Simon--Yau and Schoen--Simon theorems to 6-dimensional stable minimal immersions.
Study shows Julia sets and gasket limit sets are quasiconformally different.
problem Quasiconformal non-equivalence of Julia sets and gasket limit sets.
method Proved quasiconformal non-equivalence of Julia sets and gasket limit sets.
result Julia sets and gasket limit sets are quasiconformally different.
New deep learning model for matching sets of items, preserving exchangeability.
problem Matching two different sets of items while preserving exchangeability.
method Exchangeable deep neural networks architecture and efficient training framework.
result Significant improvements in fashion set recommendation and group re-identification.
Study shows non-symmetric convex sets have full boundary limits.
problem Understanding boundaries of non-symmetric convex sets.
method Proved using proximal limit set analysis.
result Proximal limit set equals full projective boundary for non-symmetric irreducible divisible convex sets.
The paper analyzes set-to-set matching with neural networks, focusing on theoretical generalization.
problem Theoretical analysis of set-to-set matching with neural networks.
method Generalization error analysis of set-to-set matching with neural networks.
result Theoretical insights into the behavior of set-to-set matching models.
Generative model learns to autoencode and generate sets of images.
problem Learning to represent and generate sets of images with unknown number of sets.
method Set Distribution Networks (SDNs) learn set encoder, discriminator, generator, and prior.
result SDNs can reconstruct and generate sets of images with preserved attributes.
Study on cold and freezing sets in digital images.
problem Properties of cold sets in digital images.
method Analysis of properties and relationships between cold and freezing sets.
result Examined relationships between cold and freezing sets.
Paper solves whether zero sets are mapping degree sets.
problem Whether finite sets containing zero are mapping degree sets.
method Examined oriented closed connected manifolds of the same dimension.
result Affirmative answer given for both integer and rational settings.
Maps sets to probability distributions to minimize information loss.
problem Learning to map sets to probability distributions to preserve information.
method Relates set operations to probability distribution interpolations and demonstrates a preliminary solution.
result Experimental results show the effectiveness of the set embedding approach.
Unified framework for generating set-valued outputs.
problem Handling unordered set outputs with varying sizes.
method Sequential Set Generation (SSG) framework.
result SSG outperforms baseline methods in experiments.
New model predicts sets from feature vectors without discontinuity issues.
problem Discontinuity issues in predicting sets from feature vectors.
method General model that respects set structure, auto-encodes point sets, predicts bounding boxes, and attributes.
result Model successfully predicts sets from a single feature vector without discontinuity.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
Study dynamics and topology of flows near non-saddle sets or W-sets.
problem Understanding the dynamics and topology of flows near specific invariant sets.
method Cohomological relations and global properties analysis.
result Dynamical classification of surfaces and robustness of non-saddle-sets.
Bayesian optimization for set inputs using approximate set kernels.
problem Permutation-invariant optimization over sets with black-box functions.
method Developed a Bayesian optimization method with set kernel, efficient approximate set kernel, and constrained acquisition function.
result Our method outperforms other methods in numerical experiments.
The study explores mapping degree sets and their properties for manifolds.
problem Understanding the structure and properties of mapping degree sets for manifolds.
method Analyzes the properties of mapping degree sets and their relationships with self-mapping degree sets.
result Not every multiplicative set containing 0,1 is a self-mapping degree set.
This paper studies the geometry of minimum-volume confidence sets for multinomial parameters.
problem Determining if minimum-volume confidence sets for multinomial outcomes are disjoint.
method Enumerating and covering the continuous regions of the exact p-value function to study the geometry of minimum-volume confidence sets.
result The geometry of minimum-volume confidence sets for multinomial parameters is studied, providing insights into their structure and properties.
Model learns set representations through optimized permutations.
problem Challenges in learning set representations due to permutation-invariance.
method Proposes a Permutation-Optimisation module to learn set permutations.
result Achieves state-of-the-art results on various set learning tasks.
Deep Sets approximates functions on sets with high-dimensional latent space.
problem Modeling functions of sets (permutation-invariant functions).
method Deep Sets, a method known to be a universal approximator for continuous set functions.
result Deep Sets' universal approximation property is only guaranteed with a sufficiently high-dimensional latent space.
Study online learning with set-valued feedback, showing differences between deterministic and randomized approaches.
problem Online learning with set-valued feedback, where labels are sets rather than single labels.
method Introduced new combinatorial dimensions (Set Littlestone and Measure Shattering) to characterize learnability.
result Characterized deterministic and randomized online learnability, and established bounds for various learning settings.
A stability-based method selects the most desirable conformal prediction set.
problem Selecting the most desirable conformal prediction set from multiple valid sets invalidates coverage guarantees.
method A stability-based approach that ensures coverage for the selected prediction set.
result The stability-based approach maintains coverage guarantees for the selected prediction set.
The paper links set cuspidality to function regularity and flatness.
problem Linking set cuspidality to function regularity and flatness.
method Analyzes arc-smooth functions and their properties on various sets.
result Establishes a precise link between set cuspidality and function regularity.
This work establishes properties on diffeological structures for set-valued maps and measures.
problem Establish rigorous properties on diffeological structures for set-valued maps and measures.
method Using diffeologies, the authors link various structures including set-valued maps, relations, gradients, measures, and shape analysis.
result Established rigorous properties on sample diffeologies.
MAGIC generates image collages from set templates using attention and set representations.
problem Generating image collages from set templates is challenging for classical models.
method Memory Attentive Generation of Image Collages (MAGIC) using Set-Transformer layers and set-pooling.
result MAGIC can generate image collages from set templates in one forward pass.
Causal Set Theory's Hauptvermutung is resolved in two ways, one of which is true.
problem Formulating and resolving the Hauptvermutung in Causal Set Theory.
method Two mathematically well-defined formulations of the Hauptvermutung, one of which is true.
result The Hauptvermutung is true when finite sets are replaced by countable sets.
The paper explores the equivalence of self-affine sets using a new pseudo-norm.
problem Tackles the equivalence of self-affine sets using a novel approach.
method Constructs a hyperbolic graph and uses a pseudo-norm to prove equivalence.
result Two totally disconnected integral self-affine sets are Lipschitz equivalent if and only if they have the same w-Hausdorff dimension. New methods for signal reconstruction using guiding sets and frame-less pathways.
problem Signal reconstruction in Hilbert spaces with specified properties.
method Axiomatic approach involving sample consistent and guiding sets, with reconstruction set defined as a shortest pathway.
result Existence and uniqueness of reconstruction set in Hilbert space, with derived stability and error bounds.
DMPS learns set data by connecting graph and set learning.
problem Lack of relational learning for set data.
method Deep Message Passing on Sets (DMPS) that connects graph and set learning.
result DMPS achieves competitive or superior results on synthetic and real-world datasets.
We adapted the Covertype data set for unsupervised learning.
problem Lack of suitable unsupervised learning data sets.
method Transformed the Covertype data set into the Wilderness Area data set.
result The Wilderness Area data set is more suitable for unsupervised learning.
Find limiting sets for digital cones and suspensions.
problem Digital topology cone and suspension constructions.
method Identify (m, n)-limiting sets, especially (0, 0)-freezing sets.
result Discover (0, 0)-limiting sets for digital cones and suspensions.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
Study freezing sets for digital images in a 2D grid.
problem Determine minimal freezing sets for digital images.
method Prove methods to obtain freezing sets for digital images (X, c_i) where X is a subset of Z^2.
result Examples show how methods can lead to the determination of minimal freezing sets.
Fuzzy prediction sets generalize binary predictions to include elements at varying confidence levels.
problem Binary prediction sets are limited; fuzzy prediction sets offer richer guarantees.
method Generalize prediction sets to fuzzy sets, showing they are e-values with merging properties.
result Optimal e-values lead to optimal fuzzy prediction sets, including optimal conformal prediction.
Study fixed points in digital images, introducing new invariants.
problem Understanding properties of digital images through fixed points.
method Introduce new invariants and freezing/cold sets to analyze fixed point sets.
result Existence of fixed point sets restricts maps on their complements.