New criteria for non-isometric group actions in metric spaces.
problem Understanding group actions on non-isometric spaces.
method Generalizing results from isometric to continuous group actions.
result Criterion for cocompact cyclic groups to be inessential.
We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…
Small generators found for cocompact arithmetic Fuchsian groups.
problem Determining generators for Fuchsian groups.
method Dynamical approach to construct fundamental regions.
result Set of small generators determined for cocompact arithmetic Fuchsian groups.
Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.
problem Understanding Willmore surfaces and their properties.
method Applying the 3-circle theorem to analyze the second fundamental form of Willmore surfaces.
result Proves a decay estimate of the second fundamental form along the neck region.
3-manifolds with two abelian-filled spaces.
problem Finding 3-manifolds with multiple abelian-filled spaces.
method Constructing 3-manifolds with specific embedding properties.
result 3-manifolds with at least two abelian-filled spaces.
New method R-LOCO improves local feature importance analysis.
problem Local attribution methods fail to accurately identify important features.
method R-LOCO segments input space into regions and applies global methods within.
result R-LOCO delivers more accurate local attributions.
The paper proves isoperimetric regions in specific manifolds cannot escape to infinity.
problem Exploring isoperimetric regions in asymptotically hyperbolic manifolds.
method Analyzing scalar curvature and Hawking mass to prove region exhaustivity.
result Isoperimetric regions with scalar curvature ≥ -6 cannot escape to infinity.
This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…
Unique embeddings found for 3-manifolds in S4 with abelian fundamental groups.
problem Embeddings of 3-manifolds in S4 with specific fundamental group properties. method Analyzing the fundamental groups and using homology handles to determine embeddings.
result Essentially unique embeddings for homology handles with perfect commutator subgroups.
New method learns dynamic brain communication patterns across regions.
problem Current methods struggle with time-varying brain communications and scalability.
method Adaptive Delay Model (ADM) using Markovian Gaussian Processes.
result Captures dynamic neural communication patterns over time.
Paper studies the expressivity of Convolutional Neural Networks (CNNs).
problem Understanding the expressivity of CNNs and their superiority in deep learning.
method Mathematical analysis of linear regions in one-layer and multi-layer ReLU CNNs.
result Deeper CNNs and CNNs have more expressivity per parameter than fully-connected NNs.
Core groups are link invariants defined by arc or region presentations.
problem Defining link invariants using different presentations of arcs and regions.
method Introducing core groups as link invariants defined by presentations involving arcs or regions, and extending these to virtual link diagrams.
result Properties of core groups and their extensions to virtual link diagrams are discussed.
Data science reveals unexpected tax manipulation patterns in Italian regions.
problem Detecting tax income manipulation in Italian municipalities.
method Adopting Benford's first digit law to analyze tax data.
result Marked disparities found in tax distributions across regions, contrary to expectations.
Study embeddings of 3-manifolds in S4 with nilpotent fundamental groups.
problem Embeddings of 3-manifolds in S4 with specific properties of fundamental groups. method Analyzes embeddings with nilpotent fundamental groups, determines groups with Hirsch length ≤5.
result Identifies all nilpotent groups with Hirsch length ≤5 and torsion-free.
The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.
problem Constructing reliable confidence regions for band-limited functions from noisy data.
method Improved norm bounds using Hoeffding's inequality and empirical Bernstein bound, majority voting to aggregate intervals.
result Confidence intervals retain their simultaneous coverage guarantee even when aggregated from random subsamples.
We show that if L is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of L are all in the interval (−δ,δ) for a fixed δ∈[0,1) and no complimentary region of L is an interval bundle over a surface, then each bo…
sBayFDNN bridges deep learning and functional data analysis for complex, structured data.
problem Challenges in functional data analysis, especially for complex, continuously structured data.
method Sparse Bayesian functional deep neural network (sBayFDNN) that learns adaptive functional embeddings and interpretable region selection.
result First theoretical guarantees for a Bayesian deep functional model, ensuring reliability and statistical rigor.
The paper studies how Kleinian groups can be deformed while preserving their peripheral structures.
problem Determining the extent of the island of discrete representations around the identity map in the space of representations.
method By cutting up the conformal boundary of a hyperbolic 3-manifold into a fundamental domain, the paper provides a computable region within which the fundamental domain is valid, ensuring peripheral structures remain similar under small deformations.
result The paper identifies a region in the space of representations of the fundamental group of a geometrically finite manifold, showing that groups in this region have peripheral structures that look coarsely similar.
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
Algorithm improves search efficiency for sparse signals using region sensing.
problem Efficiently search for sparse signals in large spaces.
method Greedy maximization of information gain using noisy average region measurements.
result Requires fewer measurements to recover signal locations compared to passive methods.
A new method matches similar regions in non-rigid shapes using spectra of differential operators.
problem Evaluating similarity of non-rigid shapes with partiality.
method Alignment of spectra of differential operators (SI-LBO and regular LBO) on a manifold with multiple metrics.
result Matching spectra outperforms competing methods on standard benchmarks.
The paper analyzes deep neural network classification regions and their decision boundaries.
problem Understanding the geometric properties of deep neural network classifiers.
method Empirical investigation of deep neural networks' classification regions and decision boundaries.
result Deep neural networks learn connected classification regions with flat decision boundaries.
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
Paper studies fundamental limits of communication in distributed learning.
problem Communication efficiency in model aggregation for distributed learning.
method Rate-Distortion approach to model aggregation as a vector Gaussian CEO problem.
result Derives rate region bound and sum-rate-distortion function for model aggregation.
An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let G be a Lie group acting on a space X. We show that the space of isotopy-equivalence classes of (G,X)-structures …
The existence or non-existence of Einstein metrics on 4-manifolds with non-trivial fundamental group and the relation with the underlying differential structure are analyzed. For most points (n,m) in a large region of the integer lattice, the manifold nCP2#mCP2 is sh…
SLS optimizes minimum-volume regions for conditional quantiles, bypassing density estimation.
problem Constructing minimum-volume prediction regions that satisfy conditional coverage.
method Super-level-set regression (SLS) directly optimizes geometric boundaries of conditional level sets.
result SLS optimizes regions directly, capturing complex conditional structures end-to-end.
New approach synthesizes Gaussian trees with unrecoverable correlation signs using information theory.
problem Quantifying and utilizing unrecoverable correlation signs in latent Gaussian trees.
method Information-theoretic approach, modeling as a communication channel, layered encoding framework.
result Achievable rate region for synthesizing observed data with unrecoverable signs.
Model explains how stablecoin runs are influenced by large sales and reserve quality.
problem Understanding and predicting stablecoin runs due to large sales and poor reserve quality.
method Global game model addressing both large sales and poor reserve quality, analyzing risk components.
result The probability of a run increases with large sales and decreases with precise public knowledge, but increases with precise private signals when fundamentals are weak.
The paper presents fundamental groups of complements of shadows in 4-balls.
problem Understanding fundamental groups of 4-manifold complements.
method Similar to Wirtinger presentation, focusing on contractible shadows.
result A presentation of fundamental groups of subpolyhedra in 4-balls.
The implied volatility surface (IVS) is a fundamental building block in computational finance. We provide a survey of methodologies for constructing such surfaces. We also discuss various topics which can influence the successful construction of IVS in practice: arbitrage-free conditions in both strike and time, how to…
Proposes a sensitivity framework to handle limited overlap in causal inference.
problem Limited overlap between treated and control groups in observational studies.
method Sensitivity framework based on worst-case confidence bounds on bias introduced by trimming.
result Protects against spurious findings by quantifying uncertainty in regions with limited overlap.
Proposes a method to solve deep neural networks' local minimum problem.
problem Local minimum problem in deep neural networks training.
method Transforms cross-entropy loss into risk-averse error criterion, adjusts RSI, and uses convexity region.
result Trained deep learning machine is expected to be inside a global minimum's attraction basin.
Effective risk control must make a tradeoff between the microprudential risk of exogenous shocks to individual institutions and the macroprudential risks caused by their systemic interactions. We investigate a simple dynamical model for understanding this tradeoff, consisting of a bank with a leverage target and an unl…
Researchers construct irreducible 4-manifolds with specific properties.
problem Creating smooth manifolds with specific topological and geometric properties.
method Equivariant fiber sums of Lefschetz fibrations and symplectic manifolds.
result Constructed irreducible manifolds with even intersection forms and specific topological invariants.
Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P\to Σ, over a compact oriented surface, Σ, carries a Poisson structure. If we trivialize P over a finite number of points on the boundary of Σ, then the moduli space carries a q…
Gradient estimate for linearized translator equation in R^4.
problem Analyzing singularity models of mean curvature flow in R^4.
method Proving a gradient estimate for the variation field W in the tip region.
result Sharp bound for the derivative of the variation field W in the tip region.
New approach uses satellite imagery to estimate economic growth.
problem Lack of reliable economic data in developing countries.
method Dynamic network and representation learning.
result Accurately predicts spatial gross economic expenditures.
Theorem proves topological censorship for universes with positive cosmological constant.
problem Proving topological censorship for spacetimes with positive cosmological constant.
method Developed a new theorem assuming eventual isolation of black hole collections.
result Regions near black hole collections have trivial fundamental group.
Researchers create solutions for naked singularities in Einstein vacuum equations.
problem Constructing solutions for the interior region of naked singularities in Einstein vacuum equations.
method Novel self-similarity and study of mixed degenerate elliptic-hyperbolic PDE's.
result Gluing together interior and exterior solutions produces a naked singularity.
Biondi et al. (2012) develop an analytical model to examine the emergent dynamic properties of share market price formation over time, capable to capture important stylized facts. These latter properties prove to be sensitive to regulatory regimes for fundamental information provision, as well as to market confidence c…
LIME outperforms other explainers in identifying adversarial attack regions.
problem Evaluating explainers for detecting adversarial attacks in neural networks.
method Quantitative and qualitative investigation of three explainers on adversarial examples.
result LIME outperforms classic salience and guided backpropagation in identifying adversarial attack regions.
We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation …
The study explores how 3-manifolds embed locally flatly in S4.
problem Embedding homology 3-spheres in S4 locally flatly. method Survey of known results, focusing on Seifert fibred 3-manifolds and using 4-dimension surgery.
result Characterization of embeddings by properties of complementary regions.
The paper studies a group action on a hyperbolic space derived from a lattice Veech group.
problem Investigating the geometry of a Veech group and its extensions.
method Analyzing the fundamental group of a bundle with singular Euclidean-by-hyperbolic geometry, collapsing regions to produce a hyperbolic action.
result The Veech group's fundamental group acts on a hyperbolic space, retaining most of its geometry.
Local probabilistic models simplify Bayesian classification for complex data.
problem Complex real-world data requires simpler models than global ones.
method Establish local probabilistic models for local regions, relaxing global assumptions.
result Local probabilistic models improve classification accuracy on real-world datasets.
New bound shows variational algorithms may struggle with barren plateaus.
problem Barren plateaus in quantum loss landscapes.
method General bound on loss variance and gradient decay.
result Exponential decay of gradients in subregions of barren plateaus.
Study shows code-level optimizations significantly impact deep RL algorithms.
problem Understanding the impact of implementation details on deep RL algorithms.
method Case study on PPO and TRPO, investigating the effects of code-level optimizations.
result Code-level optimizations are crucial for performance in deep RL algorithms.