We study the topology of the inertia space of a smooth -manifold where is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differ…
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48 results for “inertia spaces”
Study of forms on inertia spaces using Grauert--Grothendieck complex.
problem Understanding basic relative forms on inertia spaces of Lie group actions.
method Use of Grauert--Grothendieck complex on differentiable spaces.
result Sheaf complex of basic relative forms is a fine resolution of Brylinski's sheaf.
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
The paper studies smooth structures on quaternionic projective spaces using inertia groups.
problem Understanding smooth structures on quaternionic projective spaces.
method Computation of inertia groups and their analogues using stable homotopy theory.
result The concordance inertia group is trivial in dimension 20 but non-trivial in higher dimensions.
Differentiable groupoids and their inertia spaces are studied with a de Rham theorem.
problem Understanding the de Rham cohomology of inertia spaces.
method Introducing differentiable stratified groupoids and proving a de Rham theorem.
result Inertia groupoids of proper Lie groupoids are locally contractible differentiable stratified groupoids.
The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
problem Classifying ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
method Maximum principle applications, classification based on causal character of rulings.
result Planes are the only cylindrical stationary surfaces. For non-cylindrical surfaces, classification depends on the causal character of the rulings.
Inertial proximal gradient algorithm shows monotonically decreasing values.
problem Optimizing functions with inertia.
method Proximal gradient algorithm with alternated inertia.
result Algorithm with alternated inertia achieves monotonically decreasing functional values.
Two classification results for stationary surfaces of least moment of inertia.
problem Classifying stationary surfaces in Euclidean space based on their energy.
method Analyzing ruled and foliated surfaces, using critical point theory.
result Classification of stationary surfaces including vector planes, elongated helicoids, and specific types of surfaces.
The paper computes inertia groups of certain high-dimensional manifolds.
problem Diffeomorphism classification of -connected, smooth, closed, oriented -manifolds.
method Surgery theory, modified surgery, and special cases of conjectures.
result Inertia groups always vanish for and certain cases of .
The paper studies homotopy inertia groups and tangential structures of manifolds.
problem Understanding the homotopy inertia groups and tangential structures of manifolds.
method Analyzing the homotopy type and cohomology of manifolds to determine homotopy inertia groups and tangential structures.
result The homotopy inertia groups of certain manifolds are shown to be trivial under specific conditions.
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
In this paper we define and study the "ghost loop orbifold" of an orbifold consisting of those loops that remain constant in the coarse moduli space of . We construct a configuration space model for the ghost loop orbifold using an idea of G. Segal. From this we exhibit the relation between the Hochschild and cy…
New insights into BNN optimization redefine latent weights as inertia.
problem Optimizing Binarized Neural Networks (BNNs) with latent weights.
method Interpreted latent weights as inertia and introduced Binary Optimizer (Bop).
result Demonstrated improved performance of Bop on CIFAR-10 and ImageNet.
Novel algorithms scale correspondence analysis to large datasets.
problem Scaling correspondence analysis to large, high-dimensional datasets.
method Interpreting CA in terms of principal inertia components and using deep neural networks for approximation.
result Maximally correlated embeddings of pairs of random variables in CA can be reliably approximated from data using deep neural networks.
Planes and spheres are the only stationary surfaces with constant Gauss curvature.
problem Finding surfaces with constant Gauss curvature that are stationary under a specific energy function.
method Proving the uniqueness of stationary surfaces by considering different curvature conditions.
result Planes and spheres are the only stationary surfaces with constant Gauss curvature.
Proves triviality of inertia groups in high-dimensional manifolds.
problem Classifying manifolds in the metastable range.
method Understanding the second extended power functor in synthetic spectra.
result Inertia groups of high-dimensional manifolds are trivial.
GeoHNN models physics laws for stable, accurate predictions.
problem Violations of physical principles in machine learning models.
method Explicitly encodes geometric priors in inertia and phase space.
result Significantly outperforms existing models in long-term stability and accuracy.
We introduce and study a new class of homotopy spheres called Farrell-Jones spheres. Using Farrell-Jones sphere we construct examples of closed negatively curved manifolds , where or , which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds, thereby giving a partial answer to a que…
Extends Euler's problem to Lorentz-Minkowski plane.
problem Finding critical points of moment of inertia in Lorentz-Minkowski space.
method Explicit solutions for stationary curves, symmetries, inversions, and energy maximization.
result Explicit solutions for stationary spacelike and timelike curves, and methods to transform between them.
Neural network HDP improves virtual inertia control for non-inductive grids.
problem Traditional virtual inertia controllers are not suitable for non-inductive grids.
method Adaptive neural network heuristic dynamic programming (HDP) for optimal control.
result The proposed HDP controller outperforms traditional controllers in virtual inertia control.
New approach limits regret in non-stationary bandits.
problem Understanding worst case regret in time-varying bandits.
method Belief inertia argument to resist new evidence after changes.
result Linear worst case regret for classical and restarting algorithms.
Study rolling of a ball over a sphere in n-dimensional space.
problem Rolling of a Chaplygin ball over a fixed sphere in R^n.
method Modifications of L+R and LR systems on Lie groups with invariant measure.
result Hamiltonization of the reduced system for a special inertia operator.
Let denote a closed -connected smoothable topological -manifold. We show that the group of concordance classes of smoothings of is isomorphic to the group of smooth homotopy spheres for or , the concordance inertia group for $…
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
Enumerating Smooth Structures on math.AT
The paper computes smooth structures on a specific product manifold.
problem Computing the number of smooth structures on a product manifold.
method Using known low-dimensional computations of stable homotopy groups of spheres, the paper determines the inertia group of the product manifold.
result The paper establishes a diffeomorphism classification of all smooth manifolds homeomorphic to for .
The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group and the closed manifold subgroup are equal in dimensions , for any…
This paper explains why Adam generalizes worse than SGD by analyzing its components.
problem Understanding why Adam generalizes worse than Stochastic Gradient Descent (SGD).
method Diffusion theoretical framework to disentangle the effects of Adaptive Learning Rate and Momentum.
result Adaptive Learning Rate helps escape saddle points but not select flat minima, while Momentum provides a drift effect to help pass through saddle points.
A connection between minimal surfaces and the two-dimensional analogues of a problem of Eulermath.DG
The paper establishes a connection between minimal surfaces and a family of stationary surfaces via inversions.
problem The problem of finding minimal surfaces and their properties.
method Using inversions, the paper establishes a one-to-one correspondence between -stationary surfaces and -stationary surfaces, focusing on -stationary surfaces which are minimal surfaces.
result The paper solves the Börling problem and provides results of uniqueness for -stationary surfaces.
The paper extends Euler's problem to hyperbolic and spherical planes.
problem Extending Euler's problem to hyperbolic and spherical planes.
method Characterizing critical points of moment of inertia energy in hyperbolic and spherical planes.
result Closed stationary curves in hyperbolic plane are circles centered at N.
Study axisymmetric surfaces in Euclidean space for energy minimization.
problem Finding surfaces in Euclidean space that minimize energy.
method Phase plane analysis and maximum principle.
result Helicoidal stationary surfaces must be rotational.
Study on well-posedness of EPDiff equations with pseudo-differential inertia.
problem Analyzing the EPDiff equations with fractional Sobolev metrics.
method Fractional order Sobolev-type metrics on diffeomorphism groups, proving well-posedness.
result Proves local and global well-posedness for EPDiff equations.
We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modelling inert investors. With a suitable scaling, we show that when the price is driven …
The study identifies cylinders in certain submanifolds of Minkowski space.
problem Characterizing submanifolds with specific geometric properties in Minkowski space.
method Analyzing Finsler submanifolds with nonnegative Ricci curvature in Minkowski space, proving cylinder properties under certain conditions.
result Submanifolds containing a line or with positive relative nullity index are cylinders under specific conditions.
The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured tha…
Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spacesstat.ML
Solves memorization in diffusion models for manifold data.
problem Memorization effect in diffusion models for manifold data.
method Inertia update at the end of empirical diffusion simulation.
result Approximates true data distribution on a manifold.
Extended surgery theory proves diffeomorphism for simply-connected 4k-manifolds.
problem Proving diffeomorphism for simply-connected 4k-manifolds.
method Construction of an extended surgery obstruction associated to a normal bordism.
result Identifies the inertia group of a (2k-1)-connected 4k-manifold.
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.
We consider an SO(4) Euler rigid body with two 'inertia momenta' coinciding. We study it from the point of view of bihamiltonian geometry. We show how to algebraically integrate it by means of the method of separation of variables.
Classical dynamical equations describing a certain version of the nonHamiltonian interaction of two rotators (Euler tops with completely degenerate inertia tensors) are considered. The simplest case is integrated. It is shown that the dynamics is almost periodic with periods depending on the initial data.
A Poincaré-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler c…
We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold . Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group an orbifold called the space of -sectors o…
Proves inequalities for convex hypersurfaces in Euclidean space.
problem Geometric inequalities for hypersurfaces in Euclidean space.
method Proves two weighted geometric inequalities involving area, volume, and mean curvature.
result Examples of convex surfaces with ratios less than the round sphere.
The paper studies a natural -dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto zero value of the SO(n-1)-momentum mapping becomes an integrable Hamiltonian system after an app…
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
The paper connects PSO and CBO methods using stochastic modeling and mean-field limits.
problem Global optimization problems with particle swarm optimization and consensus based optimization.
method Stochastic differential equations and mean-field approximation to derive macroscopic hydrodynamic equations.
result Derives mean-field approximation for PSO and links it to CBO methods.
One approach to the analysis of stochastic fluctuations in market prices is to model characteristics of investor behaviour and the complex interactions between market participants, with the aim of extracting consequences in the aggregate. This agent-based viewpoint in finance goes back at least to the work of Garman (1…
The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.
problem Understanding the structure of normal subgroups in mapping class groups of surfaces with specific subsets.
method Proves two structure theorems: purity and inertia, characterizing normal subgroups.
result Characterizes finite-type normal subgroups of mapping class groups of surfaces with Cantor subsets.
The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle . The structure group G=SO(3) is the vorticity group, and the bundle ${\cal P}=GL_+(3, R})$ is the connected component of the general linear group. The base manifold is the space of positive-defi…