New method connects compression bodies through cone manifolds.
problem Understanding how compression bodies can be transformed.
method Using cone manifold holonomy groups and standard CAT(0) space techniques. result Realized all edges in compression body graph through paths.
We find a basis for a quotient algebra of Kauffman bracket skein algebra.
problem Understanding the structure of Kauffman bracket skein algebra.
method Explicit construction of a basis using quotient and augmentation ideal.
result Induces two types of finite type invariants for links in a handle body.
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…
New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.
problem Proving diffeomorphisms extend to 4D 1-handlebodies and compression bodies.
method Classification of Heegaard splittings and Cerf's theorem on 3-ball diffeomorphisms.
result Every diffeomorphism of the positive boundary extends to the whole compression body.
New model simplifies symmetry handling in generative AI.
problem Symmetry handling in generative models for scientific tasks.
method Quotient-space diffusion models, viewing symmetry as quotient space.
result Improves performance over existing methods for molecular structure generation.
Deep RL trains a robust humanoid push-recovery policy.
problem Training robust humanoid push-recovery policies.
method Model-free Deep Reinforcement Learning.
result Policy learns robust behaviors across the entire body.
A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.
Residual neural networks improve collision prediction in planetary simulations.
problem Accurate prediction of planetary collisions in N-body simulations.
method Residual neural networks trained on collision data.
result Residual neural networks outperform existing methods in prediction accuracy and generalization.
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.
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 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 S1-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 Lp 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 λ-sausage' bodies and used extra assumption of thickness to handle non-convex inner parallel bodies. result The thick λ-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 (1,1)-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.
We investigate the perturbative aspects of Rozansky-Witten's 3d σ-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of reg…
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 λ-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 C∞ and C1 smoothness, and using homeomorphism and congruence concepts. result Determine the homeomorphism type of positively curved C∞ 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 δ-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.
New dataset for UAV gesture recognition in outdoor settings.
problem Lack of public outdoor UAV gesture recognition datasets.
method Recorded 13 gestures in an outdoor setting, created 119 HD video clips, annotated with body joints and gesture classes.
result Baseline gesture recognition performance of 91.9% using P-CNN.
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…
Study floating bodies in various space forms using a new approach.
problem Investigate floating bodies in different space forms.
method Develop a unified approach to study floating bodies in Euclidean space, Euclidean unit sphere, and hyperbolic space.
result Establish a relation between the derivative of the volume of the floating body and the floating area.
Estimates potential using exterior cones of a body to restrict maximizer.
problem Estimating potential with exterior cones of a body.
method Using exterior cones of the body to estimate the potential.
result Restricts the location of a maximizer of the potential into a smaller region.
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 n-dimensional convex bodies for n≥2. When S is a closed, orientable surface with genus g(S)≥2, we show that the automorphism group of the compression body graph CB(S) is the mapping class group. Here, vertices are compression bodies with exterior boundary S, and edges connect pairs of compression bodies where one contains the other.
New algorithms improve sampling from constrained distributions.
problem Sampling from distributions constrained to convex bodies.
method Penalized Langevin Dynamics and Underdamped Monte Carlo methods.
result Improved convergence rates for constrained sampling problems.
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.
New space of tensorial bodies defined, properties and representatives studied.
problem Characterizing convex bodies in tensor norms.
method Introduced a new space of tensorial bodies, defined a Banach-Mazur distance, and proved existence of a compact type compactum.
result Topological representatives for the space of tensorial bodies and the Banach-Mazur type compactum are given.
The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.
problem Finding minimal fillings of convex bodies.
method Analyzing integral current spaces and proving rigidity properties.
result Convex bodies are the unique minimal fillings of their boundary metrics among integral current spaces and enjoy Lipschitz-volume rigidity.
A survey of existing methods for stopping active learning (AL) reveals the needs for methods that are: more widely applicable; more aggressive in saving annotations; and more stable across changing datasets. A new method for stopping AL based on stabilizing predictions is presented that addresses these needs. Furthermo…
The hyperbolic plane is derived from a three-body problem in Euclidean space.
problem Constructing the hyperbolic plane from a three-body problem.
method Scale plus symmetry reduction of a three-body problem in Euclidean plane using Jacobi-Maupertuis metric.
result The hyperbolic plane and its geodesic flow are derived from a three-body problem.
Efficiently samples arbitrary compact bodies with polynomial complexity.
problem Uniform sampling from arbitrary compact bodies efficiently.
method Warm start algorithm under isoperimetry and volume growth condition.
result Substantial generalization of known results for convex and star-shaped bodies.
Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler…
Study on Lp affine surface areas and their inequalities for convex bodies.
problem Understanding weighted Lp affine surface areas in convex bodies. method Investigating valuations, isoperimetric inequalities, and connections to f divergences. result Established isoperimetric inequalities for weighted Lp affine surface areas. Established a new inequality for convex bodies in high dimensions.
problem Bounding the product of dual quermassintegrals of convex bodies and their polar sets.
method Induction on dimensions, focusing on convex bodies in \(\mathbb{R}^{n+1}\).
result Proved a generalised Blaschke-Santalò inequality with upper bounds.
New MMD estimators detect differences in missing paired data.
problem Handling missing data in matched pairs with complex distributions.
method Maximum mean discrepancy (MMD) estimators for complex data with missing values.
result Valid and consistent estimators detect differences in data distributions.
New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.
problem Periodic solutions of the 2n-body problem and their braid types.
method Analyzing braid types and stretch factors associated with pseudo-Anosov braids.
result Braids from new periodic solutions are of pseudo-Anosov type with stretch factors as metallic ratios.