Strict concavity proven for growth indicator function of certain groups.
problem Proving strict concavity of growth indicator function for specific groups.
method Smoothness of Manhattan hypersurface and critical-exponent map.
result Strict concavity of growth indicator function for relatively Anosov groups.
The paper extends the Manhattan curve concept to complex dynamics and studies its relation to multiplier spectra.
problem Understanding the growth rate of lengths of closed geodesics in complex dynamics.
method Defining and studying the Manhattan curve for holomorphic endomorphisms of CPk and relating it to multiplier spectra. result The Manhattan curve for two holomorphic endomorphisms is related to the correlation number of their multiplier spectra.
The Manhattan curve connects metrics of hyperbolic groups, showing rigidity.
problem Understanding the relationship between different metrics on hyperbolic groups.
method Ergodic theory of topological flows and analysis of Patterson-Sullivan measures.
result The Manhattan curve is a straight line if and only if metrics are roughly similar.
In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and the convexity of Manhattan curves of the finite area type-preserving Fuchsian represe…
This study uses Twitter to analyze traveler behavior in Manhattan.
problem Analyzing traveler behavior using social media data.
method Systematic method to extract displacement information from geo-tagged tweets.
result Twitter reveals unique demographics and travel behavior patterns.
A method for camera calibration using heatmap regression for fisheye images.
problem Accurate and robust camera angle estimation from fisheye images in the Manhattan world.
method Heatmap regression to detect directions of labeled image coordinates, simultaneous rotation and fisheye distortion recovery.
result Our method outperforms conventional methods on large-scale datasets and with off-the-shelf cameras.
Develops correlation number for specific potentials and Hitchin representations.
problem Analyzing correlation numbers for potentials with entropy gaps and Hitchin representations.
method Defines a correlation number for pairs of cusped Hitchin representations and explores its connection to the Manhattan curve.
result Establishes a connection between the correlation number and the Manhattan curve, revealing rigidity properties.
Study shows exact dimensionality and regularity of manifolds for specific groups.
problem Exact dimensionality and regularity of manifolds for relatively Anosov groups.
method Dynamical methods, including finite and mixing of Bowen–Margulis–Sullivan measures.
result Manifolds are C1-regular and growth indicator is strictly concave. Paper refines cross-lingual word embeddings using Manhattan norm.
problem Sensitivity of ℓ2 norm loss function to outliers in CLWEs. method Post-processing step using ℓ1 norm to improve CLWEs. result The ℓ1 refinement substantially outperforms state-of-the-art baselines. New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.
The paper studies the correlation of Hilbert lengths for convex projective surfaces.
problem Understanding the correlation of Hilbert lengths for convex projective surfaces.
method Asymptotic formula for free homotopy classes with renormalized Hilbert length.
result The correlation number is not uniformly bounded away from zero but can be larger than a uniform strictly positive constant.
Geodesics and boundaries found for metric structures on hyperbolic groups.
problem Understanding the space of metric structures on hyperbolic groups.
method Outer automorphism invariant geodesic bicombing and boundary construction.
result Boundary contains well-known pseudo metrics and rigidity results.
Non-negative matrix factorization (NMF) approximates a non-negative matrix X by a product of two non-negative low-rank factor matrices W and H. NMF and its extensions minimize either the Kullback-Leibler divergence or the Euclidean distance between X and WTH to model the Poisson noise or the Gaussian noise.…
Finding the reduced-dimensional structure is critical to understanding complex networks. Existing approaches such as spectral clustering are applicable only when the full network is explicitly observed. In this paper, we focus on the online factorization and partition of implicit large-scale networks based on observati…
The economy globalization measure problem is discussed. Four macroeconomic indices of twenty among the "richest" countries are examined. Four types of "distances" are calculated.Two types of networks are next constructed for each distance measure definition. It is shown that the globalization process can be best charac…
Using publicly available traffic camera data in New York City, we quantify time-dependent patterns in aggregate pedestrian foot traffic. These patterns exhibit repeatable diurnal behaviors that differ for weekdays and weekends but are broadly consistent across neighborhoods in the borough of Manhattan. Weekday patterns…
A Discriminative Deep Forest (DisDF) as a metric learning algorithm is proposed in the paper. It is based on the Deep Forest or gcForest proposed by Zhou and Feng and can be viewed as a gcForest modification. The case of the fully supervised learning is studied when the class labels of individual training examples are …
In this paper we introduce three methods for re-scaling data sets aiming at improving the likelihood of clustering validity indexes to return the true number of spherical Gaussian clusters with additional noise features. Our method obtains feature re-scaling factors taking into account the structure of a given data set…
Study analyzes Airbnb booking lead times during global crises using a new metric.
problem Disruptions in booking behaviors during global crises affect forecasting accuracy.
method Normalized L1 (Manhattan) distance to assess lead time divergences.
result Identified two-phase disruption: abrupt change at pandemic onset followed by partial recovery.
The paper estimates key metrics for linear models with Markov or hidden Markov sources.
problem Estimating free energy, mutual information, and MMSE for linear models with specific signal priors.
method Replica analysis in statistical physics, focusing on Markov and hidden Markov sources.
result The linear model with Markov or hidden Markov sources can be simplified into decoupled AWGN channels.
The rise in popularity of major social media platforms have enabled people to share photos and textual information about their daily life. One of the popular topics about which information is shared is food. Since a lot of media about food are attributed to particular locations and restaurants, information like spatio-…
Generative modeling on metric graphs using neural optimal transport
problem Deep generative modeling for continuous probability distributions on metric graphs
method Embedding graph into smooth ambient space, solving entropic Kantorovich problem, projecting back onto graph
result Generator is graph-supported
The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.
problem Classifying hypersurfaces in Nil^4.
method Using Lie group structure and Codazzi conditions.
result Characterization and classification of minimal hypersurfaces in Nil^4.
Classification of hypersurfaces in homogeneous spaces with specific properties.
problem Classifying hypersurfaces in Riemannian homogeneous spaces with additional assumptions.
method Analyzing hypersurfaces under various conditions in homogeneous spaces CP3. result All extrinsically homogeneous hypersurfaces are classified in all homogeneous CP3 spaces. Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms Nn+1(c), proving properties and finding specific examples. result Proves that biconservative hypersurfaces with constant scalar curvature in N4(c) have constant mean curvature, and in N5(c), they are either rotational or constant mean curvature. The paper classifies various types of hypersurfaces in a product space.
problem Classifying hypersurfaces in a specific product space.
method Analyzing hypersurfaces with constant curvatures, product angle functions, and additional conditions.
result Different types of hypersurfaces are classified based on their properties.
The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.
problem Characterizing and understanding Laguerre isotropic hypersurfaces.
method Analyzing hypersurfaces with zero Laguerre form and constant eigenvalues of the Laguerre tensor.
result For L-isotropic hypersurfaces, if they are also L-isoparametric, the constant λ must be zero. Classifies hypersurfaces with constant isotropic curvature in space forms.
problem Identifying hypersurfaces with constant isotropic curvature in space forms.
method Analyzing complete orientable hypersurfaces and their properties.
result Hypersurfaces have constant mean curvature only if they are isoparametric, and are minimal under specific conditions.
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area m…
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
New tensors capture intrinsic embedding data of conformal hypersurfaces.
problem Classifying hypersurface invariants in conformal manifolds.
method Constructing curvatures and conformal fundamental forms.
result Finite family of tensors captures extrinsic embedding data.
The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.
problem Characterizing canal hypersurfaces in Euclidean spaces.
method Analyzing canal hypersurfaces in Euclidean n-space, focusing on E4, computing curvature properties, and proving specific cases.
result Flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones, and minimal canal hypersurfaces are only generalized catenoids.
Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
problem Understanding the relationship between compact proper Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
method Survey of existing results and related developments.
result Progress on Cecil and Ryan's conjecture on compact proper Dupin hypersurfaces.
Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.
problem Investigating Dirac operators on hypersurfaces with degenerate metrics.
method Spinorial Gauss formula, investigation of Dirac operator, relation with Riemannian curvatures.
result Established relation between Dirac operators and curvatures of the manifold and hypersurface.
In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. The study of these hypersurfaces is motivated by the fact that for indefinite almost Hermitian manifolds this class of lightlike hypersurfaces does not exist. We also establish that radical transversal…
In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…
Study confirms fractional norms and quasinorms do not help overcome curse of dimensionality.
problem Overcoming the curse of dimensionality in machine learning.
method Systematic testing of fractional norms and quasinorms (p<1) on classification problems.
result Distance concentration behavior is qualitatively the same for all norms and quasinorms as dimensionality increases.
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in E4. First, we deal with δ(2)-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of G…
Study on minimal hypersurfaces in a special normed space.
problem Characterizing minimal hypersurfaces in a specific normed space.
method Investigate translation and separable minimal hypersurfaces.
result New insights into the properties of minimal hypersurfaces.
The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
Classifies and describes hypersurfaces in Siklos spacetimes.
problem Characterizing hypersurfaces in Siklos spacetimes.
method Classification and description of totally geodesic and parallel hypersurfaces.
result A large class of minimal hypersurfaces is described.
Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n-dimensional mean convex λ-hypersurfaces are convex if λ≤0. Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
Minimal hypersurfaces are the only H-tensional in 4D space forms.
problem Classifying H-tensional hypersurfaces in 4D space forms. method Investigation of H-tensional hypersurfaces in 4-dimensional space forms of constant sectional curvature. result Minimal hypersurfaces are the only H-tensional hypersurfaces in 4D space forms. Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
Paper characterizes a special hypersurface in 5D sphere.
problem Characterizing minimal hypersurfaces in S5. method Analyzes hypersurfaces satisfying a specific curvature condition.
result Closed minimal hypersurfaces in S5 satisfying a certain curvature condition are either totally geodesic or congruent to the Cartan minimal hypersurface. The problem of determining the {\it Bonnet hypersurfaces in} Rn+1, for n>1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/o…