Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
Constructs foliations for 3-manifolds with positive scalar curvature.
problem Finding surfaces in 3-manifolds with positive scalar curvature.
method Singular foliations of compact three-manifolds with controlled properties.
result Extends Urysohn and Gromov-Lawson waist inequalities.
Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S)>= Area(W), provided the intersections of S with horizontal pla…
Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.
problem Proving uniform waist inequalities for manifolds with specific group properties.
method Using finite covers and Cheeger inequality for manifolds with Kazhdan fundamental groups.
result Finite covers of manifolds with Kazhdan groups satisfy uniform waist inequalities in codimension two.
We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms o…
3-manifolds with positive scalar curvature have controlled foliations.
problem Understanding foliations in 3-manifolds with positive scalar curvature.
method Showed a singular foliation by surfaces with controlled area and diameter.
result 3-manifolds with positive scalar curvature admit controlled foliations.
We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist …
We study the incompressible surfaces in the exterior of a cable knot and use this to compute the representativity and waist of most cable knots.
The waist size of a cusp in an orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve generated by a parabolic isometry in the maximal cusp boundary. Previously, it was shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement. In …
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
This paper presents connections between Gromov's work on isoperimetry of waists and Milman's work on the M-ellipsoid of a convex body. It is proven that any convex body K⊆Rn has a linear image K~⊆Rn of volume one satisfying the following waist inequality: Any continu…
Continuous sweepouts cover manifolds with bounded curve lengths.
problem Covering closed Riemannian manifolds with bounded curve lengths.
method Continuous family of 1-cycles parametrized by a sphere, with length bounds in terms of volume and dimension.
result Polyhedral 1-waist equals filling radius up to a constant factor.
Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.
problem Geodesics behavior on neck-degenerate manifolds with cuspidal singularities.
method Detailed multiscale analysis, blow-up techniques.
result Geodesics exhibit focussing and winding behavior as the neck degenerates.
The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded (n-q)-dimensional volume.
Minimal submanifolds in octonionic hyperbolic spaces have large volume.
problem Characterizing minimal submanifolds in locally symmetric spaces.
method Analyzing higher expansion properties and volume constraints.
result Codimension two minimal submanifolds have at least linear volume in the ambient space.
The study quantifies topological expansion properties of complexes and their embeddings.
problem Understanding topological expansion properties of simplicial complexes.
method Quantifying topological expansion through sublinear functions and proving monotonicity under regular maps.
result Proves topological expanders contain graphical expanders and gives lower bounds for specific embeddings.
Hyperbolic spaces have many cells in their fibers.
problem Finding lower bounds on the topological complexity of fibered spaces
method Using a freedom theorem for ideals in group rings of hyperbolic groups
result For large injectivity radii, there are many cells of dimension k in the fiber p^{-1}(z)
Modern machine learning algorithms are increasingly computationally demanding, requiring specialized hardware and distributed computation to achieve high performance in a reasonable time frame. Many hyperparameter search algorithms have been proposed for improving the efficiency of model selection, however their adapta…
We introduce a new critical value c∞(L) for Tonelli Lagrangians L on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that c∞(L) is strictly larger than the Mañé critical value c(L), and on every energy level e∈(c(L),c∞(L)) there exist infinitely…
New maxfaces with catenoid or planar ends constructed using node-opening technique.
problem Lack of examples of maxfaces with catenoid or planar ends.
method Adapted node-opening technique to construct maxfaces of high genus.
result Singularities on constructed maxfaces form curves around the waists of the necks, with most singularities being cuspidal edges and the rest swallowtails.
Machine learning predicts obesity causes using genetic and imaging data.
problem Predicting causes of obesity in children and adults.
method Use ML techniques like decision trees, SVM, RF, GBM, LASSO, BN, and ANN on genetic and imaging data.
result ML models accurately predict obesity causes and chronic diseases.
Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. A comparison of the performance of various machine learning models to predict the direction of a wall following robot is presented in this paper. The models were trained using an open-source dataset that contains 24 ultrasound sensors readings and the corresponding direction for each sample. This dataset was captured u…
Method predicts NAFLD risk with high accuracy and distribution-free coverage guarantees.
problem Insufficient population-level screening tools for NAFLD.
method Gradient-boosted decision trees with conformal prediction.
result Method achieves AUROC of 0.912 internally and 0.891 externally, superior to other models.
NeuroPaint infers missing brain area dynamics from multi-animal datasets.
problem Leveraging multi-animal datasets to understand interactions between brain areas.
method Masked autoencoding approach trained across animals with partial observations.
result Models can successfully reconstruct dynamics of unrecorded brain areas.
In this paper, we introduce the Lp geominimal surface area for all −n=p<1, which extends the classical geominimal surface area (p=1) by Petty and the Lp geominimal surface area by Lutwak (p>1). Our extension of the Lp geominimal surface area is motivated by recent work on the extension of the Lp a…
Overview of affine surface area and its history.
problem None explicitly stated; focuses on overview.
method None explicitly stated; focuses on overview.
result None explicitly stated; focuses on overview.
Hasse principle applied to area-minimizing submanifolds across different homology types.
problem Understanding the behavior of area-minimizing submanifolds in various homology contexts.
method Extending the Hasse principle from number theory to geometric variational problems.
result Recovering information about area-minimizing submanifolds in integral homology from those in real and mod n homology. Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established.
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …
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.
Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
Study area-minimizing subgraphs in integer lattices.
problem Finding the most efficient subgraphs in integer lattices.
method Formulated functions of bounded variations, classified subgraphs in 2D, proved properties in higher dimensions.
result Classified area-minimizing subgraphs in 2D integer lattice up to isomorphisms.
Existing attention mechanisms are trained to attend to individual items in a collection (the memory) with a predefined, fixed granularity, e.g., a word token or an image grid. We propose area attention: a way to attend to areas in the memory, where each area contains a group of items that are structurally adjacent, e.g…
Minimal surfaces in hyperbolic space have a renormalized area criterion.
problem Minimal surfaces in hyperbolic space
method Renormalized area criterion
result Y must be a totally geodesic disk
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
Study shows area-minimizing submanifolds are mostly rough, not smooth.
problem Understanding the smoothness of area-minimizing submanifolds.
method Proved non-smoothness by contradiction and established Hausdorff dimension bounds.
result Area-minimizing submanifolds are not generically smooth, resolving a conjecture.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
Minimal area of spun trefoil knot is found in 4D cubical space.
problem Finding the minimum area for a spun trefoil knot in 4D cubical space.
method Defined minimal area of cubical 2-knots, used isotopy to find minimum area for spun trefoil.
result Spun trefoil knot requires a specific minimal area in 4D cubical space.
We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.
We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.
The paper connects least area surfaces to quasi-normal surfaces in 3-manifolds.
problem Understanding the properties of least area surfaces in 3-manifolds.
method Introducing quasi-normal surfaces and showing their relationship to least area surfaces in fine triangulations.
result Least area surfaces in 3-manifolds are quasi-normal with respect to fine triangulations, and this quasi-normality leads to piecewise flat approximations.
Pedal curves derived from ellipses are invariant in area.
problem Finding invariant areas of pedal curves derived from ellipses.
method Analytical proof and explicit area expressions.
result Pedal curves derived from ellipses are invariant in area.
We prove the existence of a continuous BV minimizer with C0 boundary value for the p-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from BV functions to vector-valued measures. Our main purpose is to study the first and second v…
Contracts for Difference (CfDs) are forwards on the spread between an area price and the system price. Together with the system price forwards, these products are used to hedge the area price risk in the Nordic electricity market. The CfDs are typically available for the next two months, three quarters and three years.…
Random forests and LASSO methods improve small area estimation using auxiliary data.
problem Estimating household consumption in small areas with limited sampled data.
method Model-based small area estimation using random forests and LASSO with auxiliary information.
result Bayesian shrinkage performed best in terms of bias, MSE, and prediction interval coverages.
Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.
problem Finding surface area of arbitrary submanifolds in R^n.
method Defining natural projected areas and volumes, deriving a recursive formula.
result Derived a new surface area formula that coincides with Crofton's and De Jong's formulas.