Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
Solves Inverse Invariant Theory for sphere partitions.
problem Inverse Invariant Theory for manifold submetries of the round sphere.
method One-to-one correspondence between manifold submetries and maximal Laplacian algebras.
result Solves the Inverse Invariant Theory problem for manifold submetries of the round sphere.
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show …
New findings on mesh group-planes validate Signature-inverse Theorem under specific conditions.
problem Invalidity of existing inverse theorems for mesh group-planes.
method Classification of three and five point meshes, analysis of joint invariant signatures.
result Valid conditions for the Signature-inverse Theorem in mesh group-planes.
This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …
New approach uses distributed persistence for stable, parallelizable topological analysis of large point clouds.
problem Estimating the full persistence diagram of large point clouds is expensive, unstable, and not a sufficient statistic.
method Proposes distributed persistence as a new invariant, which is perfectly parallelizable, more stable, and has a rich inverse theory.
result The map from point clouds to distributed persistence invariants is a global quasi-isometry, interpolating between purely geometric and topological invariants.
Flow on curves in inversive geometry converges to loxodromics.
problem Gradient flow for curve length in inversive geometry.
method Invariant gradient flow for invariant length functional.
result Solutions exist for all time and converge to loxodromic curves.
SC-Net learns interpretable filters for inverse problems, achieving optimal convergence and super-resolution.
problem Solving ill-posed inverse problems with effective regularization and interpretability.
method SC-Net operates in the spectral domain, learning a pointwise adaptive filter function based on signal-to-noise ratio.
result SC-Net achieves optimal convergence rate and zero-shot super-resolution, matching theoretical bounds.
Study of knotoids using double branched covers and geometric tools.
problem Characterizing and distinguishing knotoids and knots.
method Using double branched covers and geometric invariants.
result Established a 1-1 correspondence between knotoids and knots with strong inversion.
Inverts operator on hyperbolic surfaces, constructing invariant distributions.
problem Constructing explicit inversion formula for X-ray normal operator.
method First, inversion formula for attenuated normal operator on Poincaré disk and closed hyperbolic surfaces. Then, explicit construction of invariant distributions.
result Explicit construction of invariant distributions with prescribed pushforward.
Study presents a method to induce a generalized neural network from joint group invariant functions.
problem Encoding rule of neural network internal data representation.
method Systematic method using joint group invariant function on data-parameter domain.
result Induces a generalized neural network and its inverse operator (ridgelet transform).
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev inva…
Method uses autoregressive models to interpret neural network representations.
problem Understanding and quantifying information preserved in neural network layers.
method Trains autoregressive models to invert model representations and estimate mutual information.
result Mutual information between inputs and network layers decreases over training.
The theory of convex risk functions has now been well established as the basis for identifying the families of risk functions that should be used in risk averse optimization problems. Despite its theoretical appeal, the implementation of a convex risk function remains difficult, as there is little guidance regarding ho…
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conf…
Introduces fine shape theory to simplify shape and antishape invariants.
problem Complexity and limitations of existing shape theories for metrizable spaces.
method Develops fine shape theory with a simple definition, aiming to supersede known shape theories.
result Fine shape theory unifies Čech cohomology and Steenrod-Sitnikov homology as invariants.
New method reconstructs hidden structures from noisy data.
problem Resurrecting hidden structures from incomplete or distorted data.
method Integrates Atiyah--Molino framework and Hantjies tensor.
result Exceptional robustness in noisy conditions with error-bounded reconstructions.
We construct an inverse system of unstable Vassiliev spectral sequences on the spaces of plumbers' knots, which model the homotopy type of the space of long knots, and show that the limit of these sequences contains the finite type invariants in their usual complexity. Utilizing the cell structure on the discriminant o…
We show that the Bruschlinsky group with the winding order is a homeomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit …
New method reveals properties of ReLU-networks via invertibility.
problem Understanding the behavior of deep neural networks.
method Deriving a theory on invertibility of ReLU-layers and exploring mechanisms affecting stability.
result Numerical results show characteristic properties of ReLU-networks.
Revisits Van Est theory for Lie groupoids using homotopy inverses.
problem Relating Lie groupoids and their Lie algebroids cohomology.
method Using the Perturbation Lemma from homological algebra to construct homotopy inverses.
result Constructs homotopy inverses to the van Est differentiation maps at the cochain level.
Paper develops efficient method for probability estimation.
problem Estimating probabilities with high efficiency.
method Adaptive Monte Carlo estimation using truncated inverse binomial sampling.
result Proposed method is orders of magnitude more efficient.
Develops equivariant grid homology for strongly invertible knots.
problem Invariants of strongly invertible knots.
method Equivariant grid diagrams and mapping cones.
result Equivariant unknotting numbers and genus bounds.
Develops statistical framework for resolving reward function ambiguity in inverse reinforcement learning.
problem Non-uniqueness of reward functions in inverse reinforcement learning.
method Entropy regularization combined with least-squares reconstruction of the reward from the soft Bellman residual.
result Least-squares reward function is unique and consistent with the expert policy.
TgAE constructs surrogates for inverse modeling with theory-guided training.
problem Creating accurate surrogates for inverse modeling with limited data.
method Theory-guided Auto-Encoder (TgAE) framework based on CNN architecture.
result TgAE surrogate achieves satisfactory accuracy and efficiency in uncertainty quantification and parameter inversion.
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
The paper constructs a multi-valued inverse of quasiregular maps and develops pull-back theory for differential forms.
problem Understanding multi-valued inverses of quasiregular maps and their properties.
method Using Almgren's framework of multi-valued maps and developing pull-back theory for differential forms.
result The multi-valued inverse is a quasiregular ω-curve with respect to a natural n-form ω. Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.
problem Evolution of hypersurfaces in spacetime.
method Weak solutions for hypersurfaces evolving along inverse spacetime mean curvature in asymptotically flat maximal initial data sets.
result Weak solution detects both future- and past-trapped apparent horizons.
The paper proves a generalized inverse function theorem for curved L∞ spaces.
problem Proving a generalized inverse function theorem for curved L∞ spaces. method Obstruction theory for L∞ homomorphisms and homotopy transfer theorem for curved L∞ algebras. result A morphism of curved L∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. New method learns time-invariant rewards from demonstrations.
problem Learning robust rewards for tasks with varying execution times.
method Model-based inverse reinforcement learning with time-invariant costs.
result Approach enables learning from misaligned demonstrations and generalizes spatially.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.
fkcompute calculates a knot invariant from a braid presentation.
problem Computing the Gukov-Manolescu invariant for complex knots and links.
method Three-phase pipeline: braid presentation search, state space encoding, R-matrix multiplication.
result fkcompute efficiently computes the invariant for knots and links up to 12 crossings.
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
Computational method approximates homology groups of compact metric spaces.
problem Computing homology groups of compact metric spaces.
method Inversely sequence of finite topological spaces, inverse limit, homeomorphic copy, strong deformation retract.
result Approximates homology groups of compact metric spaces.
Method constructs spirals with given tangents and curvatures.
problem Constructing a spiral with specified tangents and curvatures.
method Inversion of the involute of a circle to find the spiral.
result Spiral construction method using linear-fractional map.
Inverse spectral theory reveals shapes from sound.
problem Can the shape of a drum be determined by its sound?
method Inverse isospectral techniques applied to specific shapes.
result The regular n-gon can be uniquely determined by its eigenvalues.
Study on Frobenius manifold structures and inversion symmetry.
problem Understanding inversion symmetry in solutions to Witten-Dijkgraaf-Verlinde-Verlinde equations.
method Using conjugacy relations on Frobenius manifold structures and flat pencils of metrics.
result Geometric interpretation of inversion symmetry.
Study of Reeb spaces induced by generic maps, focusing on their homology groups.
problem Understanding the topological properties of Reeb spaces induced by generic maps.
method Introduced cobordism-like groups based on adjacent relations of connected components of inverse images of regular values.
result Top-dimensional homology groups of Reeb spaces do not vanish for certain generic maps.
In extension theory, in particular in dimension theory, it is frequently useful to represent a given compact metrizable space X as the limit of an inverse sequence of compact polyhedra. We are going to show that, for the purposes of extension theory, it is possible to replace such an X by a better metrizable compactum …
The paper addresses human-like decision-making in multi-agent systems using bounded risk-sensitive Markov Games.
problem Modeling human-like decision-making in multi-agent systems with risk-seeking and loss-aversion behaviors.
method Forward policy design and inverse reward learning with iterative reasoning and cumulative prospect theory.
result The proposed algorithms demonstrate both risk-averse and risk-seeking behaviors in multi-agent systems.
Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.
problem Modeling and understanding twisted Spin^c-bordism and its dual.
method Geometric construction using bundle gerbes, gerbe modules, and eta-invariants.
result Definition of a twisted anomaly map from differential twisted K-theory to differential Anderson dual of twisted Spin^c-bordism.
Study shows invariant curves in tubular origami dynamics, revealing geometric barriers to folding transitions.
problem Understanding the dynamics and geometric barriers in tubular origami structures.
method Kolmogorov--Arnold--Moser (KAM) theory and numerical simulations.
result Invariant curves persist in large module limits, providing phase-space interpretation of folding modes.
New method computes dense partial correlations with applications in graph theory and uncertainty quantification.
problem Sparse inverse covariance matrices are popular but dense solutions are overlooked.
method Derives approach based on inverse problem theory.
result New insights and approaches for model selection and data preprocessing.
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.