Support theorem proved for X-ray transform on certain non-compact manifolds.
problem Support theorem for X-ray transform on non-compact manifolds with conjugate points.
method Use of plane covers and support theorem for simple manifolds by Krishnan.
result Support theorem proved for simply connected 2-step nilpotent Lie groups and some non-homogeneous 3D manifolds.
This paper proves injectivity and support theorems for tensor fields on Riemannian manifolds.
problem Injectivity and support theorems for integral moments of m-tensor fields.
method Generalized Helgason's support theorem and used first m+1-integral moments of m-tensor fields.
result Injectivity and support theorems for integral moments of m-tensor fields.
A support theorem for the horocycle Radon transform f \to \hat{f} is a property of the form \hat{f} of compact support \rightarrow f of compact support. Here we prove a variation of this result where support (\hat{f}) is outside a fixed horocycle in hyperbolic space.
Support theorem for tensor fields on Riemannian manifolds.
problem Determining the support of tensor fields from ray transforms.
method Support theorem for transverse ray transform of rank 2 tensor fields.
result Support of tensor fields vanishes on geodesics where ray transform vanishes.
Injectivity and support theorem for X-ray transform on specific Lie groups.
problem Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
method General reduction principle for manifolds with uniformly escaping geodesics.
result Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
We study the weighted ray transform of integrating functions on a Lorentzian manifold over lightlike geodesics. We prove support theorems if the manifold and the weight are analytic.
For a convex domain D that is enclosed by the hypersurface ∂D of bounded normal curvature, we prove an angle comparison theorem for angles between ∂D and geodesic rays starting from some fixed point in D, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtai…
A geometric version of the Poincaré Lemma is established for the topological vector space of differential chains. In particular, every differential k-cycle with compact support in a contractible open subset U of a smooth n-manifold M is the boundary of a differential (k+1) -chain with compact support in U. Applications…
Constructs index class for manifolds with group action.
problem Index class construction for manifolds with group action.
method Constructs coarse index class with support condition using equivariant Dirac operator.
result Shows coarse relative index theorem and compatibility with suspension isomorphism.
Support selection and eventwise decoupling for simultaneous bets proven.
problem Optimizing expected utility for simultaneous independent events with multiple outcomes.
method Proved a support theorem for a broad class of strictly increasing strictly concave utilities, identifying the exact active support and proving independence from utility function.
result The exact active support is the eventwise union of single-event supports, independent of the utility function.
We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
The paper explores cycloids in normed planes, deriving properties of closed curves.
problem Characterizing closed cycloids in normed planes.
method Analyzing differential equations and support functions.
result Describes all closed hypocycloids and epicycloids with given cusps.
Extends Polydisk Theorem to Cartan-Hartogs domains.
problem Characterizing geodesics in Cartan-Hartogs domains.
method Applying the Polydisk Theorem to Cartan-Hartogs domains.
result Cartan-Hartogs domains inherit geodesic submanifolds from bounded symmetric domains.
The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
Supports conjecture about harmonic maps from S³ to S².
problem Tackles conjecture about harmonic maps from S³ to S².
method Uses conditions on Hessian and singular values to prove validity.
result Obtains pinching theorem for minimal hypersurfaces in the sphere.
X-ray transform on H-type groups solved, revealing function injectivity.
problem Injectivity in sub-Riemannian geometry.
method Fourier Slice Theorem adapted to H-type groups.
result Integrable functions on H-type groups are uniquely determined by their integrals over geodesics.
Paper studies a new curvature system and proves rigidity and gap theorems.
problem Extending CPE conjecture to manifolds with specific structures.
method Introduces (φ−CPE) system and proves rigidity and gap theorems. result Proves rigidity and gap theorems for (φ−CPE) solutions. In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…
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.
Equivalence of conformal maps proved in sub-Riemannian manifolds.
problem Equivalence of conformal maps between sub-Riemannian manifolds.
method Regularity theory for subelliptic p-Laplacian operators, sub-Riemannian p-harmonic coordinates, propagation of regularity.
result 1-quasiconformal maps are smooth on contact manifolds.
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L2 extension theorem for holomorphic vector bundles. New compact support differential cohomology theory with Pontryagin duality proof.
problem Developing a new mathematical framework for differential cohomology with compact support.
method Adapting Cheeger-Simons approach to introduce differential cohomology with compact support, proving functoriality, excision theorem, and using Pontryagin duality.
result Pontryagin duality for differential cohomology, showing isomorphism between ordinary differential cohomology and the smooth Pontryagin dual of compactly supported differential cohomology.
In this article, we initiate a geometric measure theoretic approach to symplectic Hodge theory. In particular, we apply one of the central results in geometric measure theory, the Federer-Fleming deformation theorem, together with the cohomology theory of normal cur- rents on a differential manifold, to establish a fun…
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Central limit theorem for Green metrics on hyperbolic groups.
problem Proving a central limit theorem for Green metrics on hyperbolic groups.
method Proving a central limit theorem for Green metrics on hyperbolic groups using probability measures and ordering elements.
result Proved a central limit theorem for Green metrics on hyperbolic groups.
We use SO(3) gauge theory to define a functor from a category of unoriented webs and foams to the category of finite-dimensional vector spaces over the field of two elements. We prove a non-vanishing theorem for this SO(3) instanton homology of webs, using Gabai's sutured manifold theory. It is hoped that the non-vanis…
We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new p…
We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using the same technique we also generalize Ledoux-Xia result to complete manifolds wit…
Researchers develop a method to infer reference measures from observed functionals.
problem Tackles the challenge of identifying or recovering a reference measure from observed functionals.
method Uses the property of law-invariant functionals defining lower or upper supporting sets in dual spaces of signed measures.
result Illustrates the methodology with examples and develops a modification for Value-at-Risk.
The paper analyzes sparse PCA for incomplete data and proves support recovery conditions.
problem Support recovery in sparse PCA with non-random missing data.
method Semidefinite relaxation of the ℓ1-regularized PCA problem. result Support of the sparse leading eigenvector can be recovered with high probability.
Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.
problem Extension of pluriharmonic functions on complex manifolds.
method Vanishing result for Bott-Chern cohomology combined with Ehrenpreis technique.
result Hartogs extension theorem for pluriharmonic functions on cohomologically (n−1)-complete manifolds. Study of spinors and tangent groupoid for index theorem.
problem Atiyah-Singer index theorem
method Construct rescaled spinor bundle, define convolution operation, incorporate symbol calculus.
result Algebra of smooth sections on tangent groupoid incorporates symbol calculus.
Let (M,g) be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of f along maximal geodesics vanish on an appropriate open subset of the space of geodesics in M. Under the assumption that the metric g is real-analytic, it is shown th…
The article simplifies conditions for dimensions supporting a rational projective plane.
problem Classify dimensions supporting a rational projective plane.
method Simplified Barge-Sullivan rational surgery realization theorem conditions combined with signature equation.
result A single quadratic residue equation determines dimension support.
Develops support theorem for analytic transforms in tomography.
problem Analytic wave front set resolution for integral transforms.
method Microlocal analysis, double fibration framework, wave packet transforms.
result Uniqueness and support theorems for analytic transforms.
Paper extends cohomology classes and holomorphic sections on subvarieties.
problem Tackles extension of cohomology classes and holomorphic sections on subvarieties.
method Uses quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions.
result Provides positive answers to questions and generalizes existing L2 extension theorems. Paper establishes comparison theorems for large-margin learning.
problem Data piling issue in high-dimension and low-sample size SVM.
method Large-margin unified machines (LUM) loss functions.
result New comparison theorems for all LUM loss functions.
The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold…
Study risk-constrained Kelly optimization for mutually exclusive outcomes, proving support invariance and developing a structured algorithm.
problem Risk-constrained Kelly optimization for mutually exclusive outcomes with explicit state prices.
method Analyzes the finite mutually exclusive outcome version of risk-constrained Kelly optimization with explicit state prices, proving support invariance and developing a structured algorithm.
result Support is invariant across CRRA parameter and drawdown-surrogate parameter in the overround regime.
We give a shorter proof of the following theorem of Kathryn Mann \cite{M}: the identity component of the group of the compactly supported Cr diffeomorphisms of Rn cannot admit a nontrivial Cp-action on S1, provided n≥2, r=n+1 and p≥2. We also give a new proof of another theorem of Mann: any…
L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic Q-subgroups associated to the strata. We prove a vanishing th…
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
The paper proves a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.
problem Proving a Hard Lefschetz Theorem for push-forwards of polarized twistor modules.
method Using Kashiwara and Kawai's theorem on Hodge structures and regular polarized twistor modules.
result Proves the Hard Lefschetz Theorem for push-forwards of polarized twistor modules.
We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.
The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.
problem Proving sphere theorems for hypersurfaces with W2,n regularity. method Extending Montiel-Ros argument and using Legendrian cycles.
result Proves existence of n-dimensional Legendrian cycles with 2n-dimensional support. Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…
We develop the fundamental theorem of asset pricing in a probability-free infinite-dimensional setup. We replace the usual assumption of a prior probability by a certain continuity property in the state variable. Probabilities enter then endogenously as full support martingale measures (instead of equivalent martingale…