We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body , without assuming any further regularity on the boundary of . Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of…
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48 results for “body regions”
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a conically bounded convex set, i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone. Results concerning existence of isoperimetric regions, the behavior of the isoperimetric pr…
The period of orbits in the restricted three-body problem depends on the enclosed region.
problem Understanding the period of orbits in the restricted three-body problem.
method Analyzing the relationship between the period and the enclosed region using the Jacobian integral.
result The period of a closed orbit is determined by the enclosed region and a function of the Jacobian integral.
Hyperelastic bodies in Riemannian manifolds can levitate due to curvature-induced forces.
problem Hyperelastic bodies in Riemannian manifolds can levitate due to curvature-induced forces.
method Numerical simulations of static solutions to a particular class of problems in hyperelastic mechanics.
result Hyperelastic bodies in Riemannian manifolds can levitate due to curvature-induced forces.
We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of the potential into a smaller closed region contained in the interior of the body, …
In this paper we consider the isoperimetric profile of convex cylinders , where is an -dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of , asymptotic to a right convex cylind…
The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The appro…
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prov…
Given a collection of N solutions of the (3+1) vacuum Einstein constraint equations which are asymptotically Euclidean, we show how to construct a new solution of the constraints which is itself asymptotically Euclidean, and which contains specified sub-regions of each of the N given solutions. This generalizes earlier…
Study proves existence of regions minimizing perimeter in specific geometric structures.
problem Existence of isoperimetric regions in sub-Finsler nilpotent groups.
method Analyzes nilpotent Lie groups with a bracket-generating distribution and asymmetric norms.
result Proves existence of minimizers of perimeter under volume constraint.
Mathematician summarizes protein geometry and mutation effects.
problem Understanding how proteins mutate and their structure-function relationship.
method Mathematical analysis of protein structures and functions, focusing on hydrogen bonds and secondary structure.
result Protein secondary structure regulates mutation by stabilizing or destabilizing regions.
Generalizing previous work by two of us, we prove the non-existence of certain stationary configurations in General Relativity having a spatial reflection symmetry across a non-compact surface disjoint from the matter region. Our results cover cases such that of two symmetrically arranged rotating bodies with anti-alig…
A new method for person recognition using cosine loss.
problem Recognizing the same identity across time and space with complicated scenes and similar appearance.
method Proposes a congenerous cosine loss to train a network for robust and representative features.
result The proposed method achieves better classification accuracy than previous state-of-the-arts.
A new framework uses pixel-based images for realistic cloth animations.
problem Creating virtual cloth deformations that closely match real clothing.
method Reinterpreting cloth deformation as a 2D pattern space and using CNNs.
result Our approach achieves realistic cloth animations without accurate body shapes.
Consider a convex domain B of space. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D' are convex domains with D bounded and the closure of D contained in D' then any minimal disk whose boundary lies in the boundary of D, can be approximated in an…
Convex hypersurfaces in curved spaces bound convex regions.
problem Characterizing convex hypersurfaces in curved spaces.
method Gauss-Codazzi equations, Schur comparison theorem, Alexandrov geometry.
result Closed convex hypersurfaces bound convex regions in curved spaces.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
Study geodesic X-ray transform and streaking artifacts on simple surfaces or spaces of constant curvature.
problem Streaking artifacts in CT images due to metal regions.
method Geodesic X-ray transform on nontrapping compact Riemannian manifolds with strictly convex boundaries.
result Streaking artifacts result from conormal singularities along common tangent geodesics.
Paper proposes a method to model health outcomes using varying-coefficients and KNN-based LASSO.
problem Modeling health outcomes like BMI and cholesterol levels with varying age effects.
method Varying-coefficients regional quantile regression via KNN fused LASSO, with ADMM algorithm.
result Efficacy in capturing complex age-dependent associations between health outcomes and risk factors.
We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of norms on admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the exist…
Constructs initial data for multiple black holes with specified ADM parameters.
problem Forming multiple black holes with specific ADM parameters.
method Smooth, asymptotically flat vacuum initial data with prescribed ADM energy, momentum, and angular momentum.
result Maximal development of data results in spacetimes containing multiple black holes.
The paper proves a conjecture about the shape of floating bodies.
problem The shape of bodies of flotation and buoyancy.
method Modern differential geometry techniques.
result If a body of flotation is homothetic to a body of buoyancy, it must be an ellipse.
The Orlicz-Petty bodiesmath.MG
The paper studies properties of Orlicz-Petty bodies and related geometric areas.
problem Properties of Orlicz-Petty bodies and related geometric areas.
method Established properties through the existence and uniform boundedness of Orlicz-Petty bodies.
result Geominimal surface areas are continuous under certain conditions on convex bodies.
New material groupoid theory subdivides non-uniform bodies into smoothly uniform parts and isolated points.
problem Lack of differentiability in material bodies leads to non-uniformity.
method Introducing material groupoid and material distribution to study non-uniform bodies rigorously.
result Material bodies can be subdivided into smoothly uniform parts and isolated points.
New index characterizes non-smooth Zoll convex bodies.
problem Characterizing non-smooth Zoll convex bodies.
method Defining systolic -index and using it to introduce generalized Zoll convex bodies.
result Generalized Zoll convex bodies coincide with classical ones under certain conditions.
Infinite diameter found in compression body graph.
problem Determining the diameter of compression body graphs.
method Analyzing the structure and connections within compression body graphs.
result The compression body graph has infinite diameter.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing relative surface areas, proving invariance and inequalities, and using geometric interpretations.
result Established inequalities and a new notion of entropy for ball-convex bodies.
The paper finds the unique minimizer of area for hyperbolic bodies with curvature constraints.
problem Finding the unique minimizer of area for hyperbolic bodies with curvature constraints.
method Introduced the concept of 'thick -sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies.
result The thick -sausage body is the unique minimizer of area among all bodies with a given length and curvature constraints.
Study shows volumes of complex classes can be represented by convex bodies.
problem Understanding volumes of complex classes on Kähler manifolds.
method Approximation by partial Okounkov bodies, restricted volume properties, and bimeromorphic behavior of currents.
result Volume of transcendental big -classes can be realized by convex bodies.
Study floating bodies of polytopes, linking volume to flags.
problem Understanding floating bodies of polytopes in various spaces.
method Introducing flag simplices to connect metric and combinatorial structures.
result Weighted volume depends on complete flags of polytopes.
Improves tensor networks for classifying medical images.
problem Classifying 2D and 3D medical images efficiently.
method Develops LoTeNet, a tensor network that treats small image regions as orderless and aggregates local representations hierarchically.
result LoTeNet achieves comparable or superior performance to other methods with less computational resources.
A unique volume minimizer is found in a class of convex bodies.
problem Finding a unique minimizer for a reverse isoperimetric problem.
method Proving a reverse quermassintegral inequality.
result The convex hull of two balls is a unique minimizer among -concave bodies.
Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.
problem Defining surface area in non-Euclidean geometries.
method Generalizes illumination bodies to Riemannian spaces of constant curvature and projective Finsler geometries, proving their volume derivative defines surface area.
result Derivative of volume of illumination bodies defines surface area in non-Euclidean geometries.
Solves Alexandrov's problem for hyperbolic convex bodies.
problem Finding a convex body with a given curvature measure in hyperbolic space.
method Defined Gauss curvature measure, proved existence and uniqueness of solution.
result Uniqueness of the solution to Alexandrov's problem in hyperbolic space.
Study of smooth convex bodies up to congruence.
problem Understanding hyperspaces of smooth convex bodies up to congruence.
method Systematic study of hyperspaces of convex bodies, focusing on and smoothness, and using homeomorphism and congruence concepts.
result Determine the homeomorphism type of positively curved convex bodies and their quotient by isometries.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
Small bodies follow geodesics in general relativity.
problem Understanding the motion of small bodies in space-time.
method Analyzes the motion of small bodies as timelike geodesics or Lorentz-force curves in general relativity.
result Clarifies the relationship between modeling bodies as distributions or smooth fields.
Machine learning shapes bodies via inverse scattering.
problem Generating 3D human bodies with customizable characteristics.
method Combining inverse scattering with machine learning.
result Generates geometric bodies from characteristic parameters.
Study on discrete Okounkov bodies and their applications.
problem Understanding stability and thresholds in higher dimensions.
method Analysis of discrete Okounkov bodies and gap phenomena.
result Asymptotic analysis of stability and thresholds.
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of resu…
Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
problem Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
method Define the -illumination body and prove a generalization of Werner's formula.
result Prove a generalization of Werner's formula for the volume of illumination bodies on Riemannian manifolds.
For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
This paper solves deep learning's edge sensitivity issue by swapping important and irrelevant segments in synthetic data.
problem Edge sensitivity and high computational cost in deep learning classification models.
method Synthetic data with swapped segments to implicitly define receptive fields, preserving label information.
result The method drives networks to early convergence and appropriate solutions, improving person re-identification.
The paper proves no multiple equichordal points exist in convex bodies.
problem Existence of multiple equichordal points in convex bodies.
method Topological tools like the Borsuk-Ulam theorem and analysis of convex body properties.
result Nonexistence of multiple equichordal points in -dimensional convex bodies for .
When is a closed, orientable surface with genus , we show that the automorphism group of the compression body graph is the mapping class group. Here, vertices are compression bodies with exterior boundary , and edges connect pairs of compression bodies where one contains the other.
iDEM generates samples from Boltzmann densities without data.
problem Generating statistically independent samples from unnormalized distributions.
method Iterative algorithm using energy and gradient for diffusion-based sampler training.
result iDEM achieves state-of-the-art performance and trains faster than existing methods.
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…
The paper extends inequalities for projection bodies to arbitrary measures.
problem Sharp bounds for volume ratios of convex bodies and their projection bodies.
method Generalizations of Zhang's inequality to arbitrary measures and extensions of the projection body operator.
result New Zhang-type inequalities for arbitrary measures and functions.