Study on cosine function in smooth normed spaces, proving symmetry and characterizing planes.
problem Understanding cosine function properties in smooth normed spaces.
method Proved symmetry and derived cosine function in terms of norm's Gateaux derivative.
result Cosine function is symmetric if and only if space is Euclidean.
Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP2. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…
Unified proof of end-point estimates for Radon transform on curved spaces.
problem Proving end-point estimates for the totally-geodesic Radon transform on spaces of constant curvature.
method Unified geometric approach to prove end-point estimates for Radon transform on spaces of constant curvature.
result Unified formula for the k-plane transform of radial functions on spaces of constant curvature. We show injectivity of the X-ray transform and the d-plane Radon transform for distributions on the n-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the n-torus for tensor fields of any order, allowing the tensor…
Solves Ricci flow on Riemann surfaces with measure initial data.
problem Existence and smoothness of Ricci flow on Riemann surfaces.
method Formulation and solution of existence problem using Ricci flow.
result New examples of nongradient expanding Ricci solitons.
Combines non-Euclidean and de Sitter geometries on the plane.
problem Exploring Penrose's Conformal Cyclic Cosmology.
method Geometric model combining Beltrami-Klein and de Sitter spaces.
result Discovery of hidden ${f G}_2$ symmetry in de Sitter spaces.
New model estimates indoor radon distribution with higher spatial resolution.
problem Accurate estimation of indoor radon concentration for health assessment.
method Quantile regression forest and probabilistic Monte Carlo sampling.
result Approximate lognormal distribution of indoor radon in Germany with specific exceedance probabilities.
Abstract: Comprehensive overview on curvatures in normed planes.
problem Lack of systematic knowledge on curvatures in normed planes.
method Systematic review and derivation of new results.
result Introduction of a new curvature type and verification of its properties.
Introduces a new space of Radon measures for better understanding persistence diagrams.
problem Lack of optimal transport-based formalism for persistence diagrams.
method Formalizes persistence diagrams as Radon measures on the upper half plane via optimal partial transport.
result Characterizes convergence and barycenters of persistence diagrams.
Holographic reconstruction over local fields using Radon transform.
problem Holographic reconstruction of quantum fields in anti-de Sitter space over local fields.
method Novel construction of Radon transform and its inverse on anti-de Sitter space over local fields.
result Holographic bulk reconstruction can be formulated as the inverse Radon transform.
We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover, if it can …
We interpret the setting for a Radon transform as a submanifold of the space of generalized functions, and compute its extrinsic curvature: it is the Hessian composed with the Radon transform.
Technical proofs for Radon-Nikodym derivative identities.
problem Formalizing and proving theorems on Radon-Nikodym derivatives.
method Careful consideration of conditional and marginal probability measures.
result New interpretation of mutual and lattum information sums.
Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for ge…
We estimate Radon-Nikodym derivatives using regularization in reproducing kernel Hilbert spaces.
problem Estimating Radon-Nikodym derivatives in various applications.
method General regularization scheme in reproducing kernel Hilbert spaces.
result High order accuracy in reconstructing Radon-Nikodym derivatives at any point.
A novel approach to interpolation, classification, and clustering using Radon-Nikodym derivatives.
problem Interpolation, classification, and clustering problems in data analysis.
method Radon-Nikodym approach with Lebesgue quadrature for optimal clustering.
result The approach changes both probabilities and the probability space with new observations.
The paper derives Pizzetti formulae and inverts the Radon transform on spheres.
problem Inverting the Radon transform on spheres.
method Obtained Pizzetti-type formulae on sphere regions, used delta distributions, and derived inversion formulae.
result Derived Pizzetti formulae and inversion formulae for the Radon transform on spheres.
Kernel estimator optimally recovers function from noisy exponential Radon transform.
problem Inverting noisy exponential Radon transform of a function.
method Proposed a kernel estimator to estimate the true function.
result The estimator converges to the true function at minimax optimal rate.
Extends SW and GSW to compare heterogeneous joint distributions.
problem Limited applicability of SW and GSW to heterogeneous joint distributions.
method Introduces HHRT and PGRT to extend SW and GSW.
result H2SW distance for heterogeneous joint distributions.
We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to S1 nor to S3. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Rado…
Paper proves uniqueness of Ricci flows from nonatomic measures on surfaces.
problem Existence and uniqueness of Ricci flows from nonatomic Radon measures.
method Combining previous work, established existence and proved uniqueness.
result Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces.
The paper explores properties of the Radon transform in relation to neural networks and ridges.
problem Understanding the Radon transform and its application to neural networks and ridges.
method Investigates properties of the Radon transform, introduces new subspaces, and characterizes ridges for any distributional profile.
result Clarifies and simplifies results on the optimality of ReLU networks using the Radon transform.
Let S be a non-exceptional oriented surface of finite type. We classify all Radon measures on the space of measured geodesic laminations for S which are invariant under the mapping class group.
A new metric HSW derived from hierarchical Radon Transform addresses computational bottlenecks in sliced Wasserstein.
problem Computational inefficiency of sliced Wasserstein in high-dimensional settings with few supports.
method Hierarchical Radon Transform (HRT) and bottleneck projections to reduce projection number.
result HSW metric derived from HRT is computationally efficient and maintains metric properties.
If G is a finite group, is a function f:G→C determined by its sums over all cosets of cyclic subgroups of G? In other words, is the Radon transform on G injective? This inverse problem is a discrete analogue of asking whether a function on a compact Lie group is determined by its integrals over all ge…
Direct proofs of implications between three theorems on maps of simplex.
problem Understanding relations between three theorems on maps of simplex.
method Direct proofs using interesting relations between van Kampen and Conway-Gordon-Sachs numbers.
result Exhibited relations and direct proofs of implications between the theorems.
We prove that a transversely equicontinuous minimal lamination on a locally compact metric space Z has a transversely invariant Radon measure. Moreover if the space Z is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling.
Artifacts appear in broken ray transform due to conjugate points.
problem Artifacts in broken ray transform due to conjugate points.
method Integral transform over broken rays, analysis of conjugate points.
result Singularities cannot be recovered from local data, leading to artifacts.
Survey on inverse exponential Radon transform methods.
problem Analytical methods for inverse exponential Radon transform.
method Derivation of classical inversion formula, finite Hilbert transform, exact reconstruction from partial measurements, diverging-beam data.
result Exact reconstruction from 180 degree data using finite Hilbert transform.
Let M be a Riemannian globally symmetric space of compact type, M′ its set of maximal flat totally geodesic tori, and ad(M) its adjoint space. We show that the kernel of the maximal flat Radon transform τ:L2(M)→L2(M′) is precisely the orthogonal complement of the image of the pullback map…
Study on porous medium equation on curved manifolds, proving existence and uniqueness of solutions.
problem Existence and uniqueness of weak solutions for the porous medium equation on negatively curved Riemannian manifolds.
method Investigation of weak solutions taking initial data as finite Radon measures, proving existence and uniqueness in nonnegative solutions.
result Existence and uniqueness of solutions for the porous medium equation on negatively curved Riemannian manifolds.
New gradient flows improve high-dimensional sampling.
problem Sampling from high-dimensional target densities.
method Introducing Radon--Wasserstein gradient flows.
result Linear scaling in particles and dimensions.
The article studies mapping properties of Radon transform and backprojection on a unit ball.
problem Polyhomogeneous mapping properties of Radon transform and backprojection operator on the unit ball.
method Constructs a double b-fibration to desingularize the point-hyperplane relation, provides formulas and sharper estimates.
result Sharper estimates on polyhomogeneous mapping properties of Radon transform and backprojection compared to classic estimates.
Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.
problem Proper actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces.
method Analysis of symmetric spaces and rigidity results.
result Any connected non-compact semisimple Lie group acting properly on these spaces must be globally isomorphic to Spin(n,1) up to compact factors. Neural networks explained through geometric projections.
problem Understanding the geometric and mathematical underpinnings of neural networks.
method Exploiting connections between integration, Radon transforms, and neural networks.
result Distribution of neural network outputs can be interpreted as nonlinear projections along hypersurfaces.
In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.
Neural networks with ReLU^k approximate Sobolev functions efficiently via Radon transform.
problem Approximating functions from Sobolev spaces using shallow ReLU^k neural networks.
method Utilizing the Radon transform and discrepancy theory, we provide nearly optimal approximation rates.
result Optimal approximation rates for smoothness up to order s = k + (d+1)/2.
We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph K7 into $\…
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
Sharp estimates for Bergman metrics derived from Kähler quantization.
problem Estimating Bergman metrics in Kähler quantization.
method Upper and lower bounds on the Bergman metric expressed in terms of φ. result Optimal C1,1ˉ-convergence for quantization of Kähler currents. A theorem proves a surface evolution graph satisfies a PDE under specific conditions.
problem Prove a surface evolution graph satisfies a PDE under specific conditions.
method Use Brakke's formulation of velocity and analyze the distributional time derivative of the graph.
result The graph satisfies the PDE pointwise under the given conditions.
The study defines divergence for multivector fields on infinite-dimensional manifolds.
problem Defining divergence for multivector fields on infinite-dimensional manifolds.
method Definition of divergence consistent with finite-dimensional geometry, properties transferred from finite to infinite dimensions.
result Natural properties of divergence are preserved in infinite dimensions.
Adapts to estimate functions from noisy ERT data.
problem Estimating functions from noisy Exponential Radon Transform data.
method Locally adaptive kernel type estimator for functions of varying smoothness.
result Achieves minimax optimal rate up to a log(n) factor for Sobolev functions.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …
There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for …
This paper reinterprets and generalizes Hurwitz--Radon numbers using Lie groups and manifolds.
problem Understanding and generalizing Hurwitz--Radon numbers in Lie algebra settings.
method Defining new numbers ρG,s(M,σ) and ρG,s±(M,σ,abla) for G-manifolds and their affine connections. result New numbers coincide with Kannaka--Tojo's ρ(1)(g,ι) and ρ(2)(g,ι) in certain cases. Extends DC Calculus to finite dimensional spaces with curvature constraints.
problem Defining Hessian and Laplacian in non-smooth spaces.
method Extending DC Calculus to Alexandrov spaces, defining Hessian and Laplacian as measure-valued tensors and Radon measures.
result Hessian and Laplacian properties in non-smooth spaces mirror those on smooth manifolds.
Study normal operators of double fibration transforms with conjugate points.
problem Normal operators of double fibration transforms with conjugate points.
method Stable conditions on the distribution of conjugate points, splitting into elliptic and Fourier integral operators.
result Normal operator splits into an elliptic pseudodifferential operator and Fourier integral operators.