The paper proves finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.
problem Proving finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.
method Using positive biRicci curvature and uniform scalar curvature bounds, the paper shows that the Urysohn 1-width is finite and depends only on the curvature bounds.
result Closed 4 and 5 manifolds with positive biRicci curvature have finite Urysohn 1-width, which depends only on the curvature bounds.
Sharp dimension constraints for positive intermediate curvature metrics are established.
problem Proving sharp dimension constraints for metrics with positive intermediate curvature.
method Constructing counterexamples and extending rigidity results.
result Sharp dimension constraints for positive intermediate curvature metrics are established.
Enhanced binary classifier uses Urysohn's Lemma of Topology.
problem Binary classification challenges.
method Utilizes Urysohn's Lemma of Topology to construct separating functions.
result Exceptional performance in numerical experiments (95% to 100%).
The paper bounds the min-max width of embedded circles on spheres and manifolds.
problem Bounding the min-max width of embedded circles on spheres and manifolds.
method Inducing a sweepout by pairs of points in embedded circles from a given sweepout of the sphere by closed curves.
result Lower bounds for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles.
The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.
problem Disproving the macroscopic version of Gromov's Urysohn width conjecture for scalar curvature.
method Novel estimate on Urysohn width of circle bundles and a new notion of ruling for Riemannian manifolds.
result The macroscopic version of Gromov's Urysohn width conjecture for scalar curvature is false in dimensions four and above.
Study bounds Urysohn width of manifolds under surgeries.
problem Bounding Urysohn width of manifolds after surgeries.
method Analyzes connected sums and universal covers, applies to general surgeries.
result Optimal constants in estimates of width bounds are shown.
The paper proves conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.
problem Conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.
method Adaptation of Guth's macroscopic version of the Schoen-Yau descent argument.
result A complete Riemannian manifold with positive macroscopic scalar curvature contains a non-nullhomologous hypersurface of small Urysohn width.
We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…
Study of manifolds with specific curvature properties using capillary surfaces.
problem Obtaining geometric properties of manifolds with nonnegative scalar curvature and strictly mean convex boundary.
method Use of stable capillary surfaces and Urysohn width to study geometric properties.
result Obtained an obstruction to filling 2-manifolds by 3-manifolds.
We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, …
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance …
We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone R of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance matrix is defined and we proved that the set of such matrices is everywhere dense …
Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Ste…
The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.
problem Existence of positive scalar curvature metrics on non-orientable manifolds and their covers.
method Extends Schoen-Yau inductive descent approach to non-orientable manifolds.
result Examples of non-orientable manifolds with positive scalar curvature metrics on their orientation double covers but not on homotopy equivalent manifolds.
New metric properties show volume constraints in collapsing spaces.
problem Volume constraints in collapsing spaces.
method Generalization of recent progress in metric geometry involving the volume of balls of radius in a certain range with collapsing at different scales.
result For every Riemannian metric on a manifold of sufficiently small volume, there is a point with volume constraints in the universal cover.
Study shows bound on Uryson width for specific 3D manifolds.
problem Bounding Uryson width for 3D manifolds with non-negative Ricci curvature and strictly mean convex boundary.
method Proved existence of a Morse function with uniform diameter bounds on level sets.
result Upper bound on Uryson width for the specified 3D manifolds.
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.
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz φ-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz φ-radial addition of two star bodies, we derive a f…
Let $f : X \lo Y$ be a map of compact metric spaces. A classical theorem of Hurewicz asserts that dimX≤dimY+dimf where dimf=sup{dimf−1(y):y∈Y}. The first author conjectured that {\em dimY+dimf in Hurewicz's theorem can be replaced by $\sup \{\dim (Y \times f^{-1}(y)): y \in Y \…
Efficiently learns Ising model parameters with limited statistics.
problem Learning Ising model parameters with limited sample configurations.
method Examines trade-offs between computation and observation, using Ising model as example.
result Reconstructs model parameters with statistics up to order O(γ) for ℓ1 width γ. We prove a new version of isoperimetric inequality: Given a positive real m, a Banach space B, a closed subset Y of metric space X and a continuous map f:Y→B with f(Y) compact infFHCm+1(F(X))≤c(m)HCm(f(Y))mm+1, where HCm denotes the m-dimensional Hausdorff content,…
The study examines manifolds with specific curvature properties and finds topological and metric constraints.
problem Analyzing manifolds with almost non-negative Ricci curvature and positive scalar curvature bounds.
method Using curvature bounds to deduce metric and topological properties.
result The manifold has at most linear volume growth and at most two ends.
A new memory system handles non-stationary environments by self-sizing and retaining memories.
problem Non-stationary environments where memories arrive over time and must be distinguished from noise.
method A self-sizing continual associative memory that generalizes Hopfield's network, handling adaptation and novelty.
result The memory system grows to the intrinsic memory demand of the environment and retains memories without forgetting.