Study on extending curves in sub-Riemannian manifolds with compatibility conditions.
problem Validating Whitney extension property for horizontal curves in sub-Riemannian manifolds.
method Analyzing equiregular and singular sub-Riemannian manifolds, using nilpotent approximation and Lusin-like approximation.
result Extension property holds for horizontal curves in sub-Riemannian manifolds with specific conditions.
New L0 norm added to TDA for market analysis.
problem Improving TDA tools for market prediction.
method Defined and applied L0 norm in TDA for four markets.
result Enhanced TDA tools for market analysis.
Differentiable sorting and rank normalization are incompatible, with specific conditions for admissibility.
problem Incompatibility between differentiable sorting and rank normalization.
method Formalized admissibility through monotone invariance, batch independence, and rank-space stability conditions.
result Different gap-sensitive and batchwise relaxations of rank normalization violate the conditions for admissibility.
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examin…
Stable saddle solutions found for specific dimensions of the Allen-Cahn equation.
problem Stability of saddle solutions for the Allen-Cahn equation in specific dimensions.
method Analyzing the Simons cone and energy functional to confirm saddle solutions' stability.
result Stable saddle solutions found for m=4,5,6. Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
We define disentanglement in generative models and prove it's related to identifiable factors.
problem Understanding disentanglement in generative models like VAEs and GANs.
method Characterized disentanglement in smooth generative pushforward models using the SVD of the Jacobian.
result Disentanglement is identifiable under certain conditions on the generator, promoting separable factors.
Any cyclic quadrilateral can fit inside any smooth curve.
problem Inscribing cyclic quadrilaterals in smooth curves.
method Proving geometric properties of cyclic quadrilaterals and smooth curves.
result Cyclic quadrilaterals can be inscribed in any closed convex C1-curve. Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category BunG,M of principal G-bundles over a given manifold M, I develop in this paper the same ideas for the…
Revises Schwarzschild manifold rigidity proof for spin manifolds.
problem Rigidity of Schwarzschild manifold for spin manifolds.
method Spinorial proof approach.
result Generalizes and includes classical and recent black hole uniqueness theorems.
Paper proves smoothness of solutions to a complex geometric problem.
problem Smoothness of solutions to the degenerate Lp Dual Minkowski problem. method Inspired by Guan and Li's approach for the Aleksandrov problem, the authors derive C1,1 estimates. result Proves solutions are C1,1 regular. Generalizes Hecke algebra for double torus, linking to skein algebra.
problem Understanding algebraic structures on double torus.
method Introducing Heegaard dual operators and Dehn twists.
result Established relationship between Hecke algebra and skein algebra.
Study optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
problem Optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
method Using Gauss and Codazzi equations, prove an optimal inequality.
result Prove an optimal inequality for contact CR-warped product submanifolds.
Let (X,L) be a polarized Kähler manifold that admits an extremal Kähler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti to ex…
A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.
problem Solving the quaternionic Monge-Ampère equation for (n−1)-quaternionic plurisubharmonic functions on a hyperKähler manifold. method Proves existence and uniqueness of solutions using a Cherrier-type inequality and C1 and C2 estimates. result Obtains smooth solutions to the quaternionic Monge-Ampère equation.
Study confirms C1 regularity for convex functionals with bounded degeneracy set.
problem Confirming C1 regularity for minimizers of convex functionals with small degeneracy set. method Building on previous work, confirms C1 regularity when D2F is positive and bounded away from finitely many points. Constructs a counterexample in R4 where F is strictly convex but D2F degenerates on a Simons cone intersection. result Confirms C1 regularity for minimizers of convex functionals with small degeneracy set. The paper proves regularity for minimal surfaces near polyhedral cones.
problem Understanding the regularity of minimal surfaces near polyhedral cones.
method Adapting Simon's method and establishing C1,α-regularity for minimal varifolds. result Proves C1,α-regularity for minimal varifolds near polyhedral cones. New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on m-Hessian operators. result Deduced an a priori C1-estimate for solutions to the Dirichlet problem for m-Hessian equations. Survey on algebraic fibers of group extensions and their finiteness properties.
problem Existence and finiteness of algebraic fibers in group extensions.
method Analysis of abstract and pro-p groups. result Finiteness properties of algebraic fibers in group extensions.
We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.
Analytic sub-Riemannian geodesics in 3D are always C1 and analytic except finitely many points.
problem Regularity of sub-Riemannian minimizing geodesics in 3D analytic manifolds.
method Investigation of totally nonholonomic analytic distributions, proof of Hausdorff dimension, and regularity of minimizing geodesics.
result Minimizing sub-Riemannian geodesics in 3D analytic manifolds are C1 and analytic except finitely many points.
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
New risk measure extensions preserve key properties.
problem Extending risk measures to larger spaces while preserving properties.
method Unique extension of dilatation monotone risk measures to L1. result Risk measures extend uniquely and preserve monotonicity, convexity, and cash-additivity.
The paper extends fractional Laplacian theory using symmetric spaces and extension problems.
problem Developing a new framework for fractional Laplacians.
method Using representation theory on symmetric spaces and extension problems.
result Constructed new boundary operators with conformal properties.
The paper studies groups with proper actions on finite products of hyperbolic spaces.
problem Characterizing groups with proper actions on finite products of hyperbolic spaces.
method Lineal actions and central extensions of groups.
result Groups with property (PH) or (PH') have specific properties related to their actions on hyperbolic spaces.
New group found not satisfying quasi-isometric triviality property.
problem Identifying groups that do not satisfy a specific quasi-isometric property.
method Analyzing central extensions and cohomology properties of groups.
result Found a finitely generated group that does not satisfy Property QITB.
This research extends quasiplurisubharmonic functions on compact Kähler manifolds.
problem Extending quasiplurisubharmonic functions on compact Kähler manifolds.
method Using a cover of Zariski-open Stein sets with strictly plurisubharmonic potentials, the authors prove extension properties for plurisubharmonic functions.
result Any ω|_X-plurisubharmonic function on an analytic subvariety X of a compact Kähler manifold V extends to a ω-plurisubharmonic function on V.
We prove extension theorems for several geometric properties such as asymptotic property C (APC), finite decomposition complexity (FDC), strict finite decomposition complexity (sFDC) which are weakenings of Gromov's finite asymptotic dimension (FAD). The context of all theorems is a finitely generated group G with a …
The paper classifies extensions of Yang-Mills-type theories, proving maximality and universality are dense properties.
problem Classifying extensions of Yang-Mills-type theories.
method Categorical characterization and dense properties analysis.
result Maximality and universality are dense properties in the one-point compactification of extension classes.
Study on finiteness property of right-angled Artin groups actions on extension graphs.
problem Finiteness property of hyperbolic simplicial actions on right-angled Artin groups.
method Analysis of right-angled Artin group actions on extension graphs, using asymptotic translation lengths and syllable lengths.
result Asymptotic translation lengths of elements in right-angled Artin groups are rational and have a common denominator under certain conditions.
We present the recent advances along with an error analysis of the IBM speaker recognition system for conversational speech. Some of the key advancements that contribute to our system include: a nearest-neighbor discriminant analysis (NDA) approach (as opposed to LDA) for intersession variability compensation in the i-…
Extends topological groupoids and studies their properties.
problem Understanding topological groupoid extensions and their properties.
method Introduces topological groupoid extensions and relates them to gerbes over topological stacks.
result Properties of gerbes over Serre, Hurewicz stacks are studied.
Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.
The properties of the Riemann extensions of nonriemannian spaces defined by the first order systems of differential equations are considered.
Study optimal holomorphic extensions on complex manifolds with transitivity property.
problem Optimal holomorphic extensions on complex manifolds with transitivity property.
method Use Toeplitz operators and transitivity property for optimal holomorphic extensions.
result Transitivity property of optimal holomorphic extensions with small defect.
In this paper we show how to relate European call and put options on multiple assets to certain convex bodies called lift zonoids. Based on this, geometric properties can be translated into economic statements and vice versa. For instance, the European call-put parity corresponds to the central symmetry property, while…
The paper extends sequences while preserving statistical properties using a mixture model.
problem Extending sequences while retaining their statistical properties.
method Auto-regressive Sequence Extension Mixture Model (SEMM) using deep learning.
result The mixture model outperforms traditional neural networks in sequence extension with statistical property retention.
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. Let $\H^n$ be the Heisenberg group of topological dimension 2n+1. We prove that if n is odd, the pair of metric spaces $(\H^n, \H^n)$ does not have the Lipschitz extension property.
Extends growth properties of hyperbolic groups to their extensions.
problem Quantifying subgroup alternatives in group laws.
method Develops a framework for preserving exponential growth in extensions of hyperbolic groups.
result Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.
The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.
problem Understanding the fundamental groups of compact Kahler varieties with specific properties.
method Application of geometric analytic theory of Kahler spaces and study of Albanese maps.
result The fundamental group of compact Kahler varieties with nef anti-canonical bundle is almost Abelian.
Let M be an n−dimensional differentiable manifold with a symmetric connection ∇ and T∗M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension on T∗M defined by means of a symmetric -tensor field c on M.…
Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.
problem Establishing conditions for optimal L2 extension in complex geometry.
method Analyzing singular Nakano positivity and applying L2 extension theorem.
result Necessary condition for equality in optimal L2 extension theorem.
It is shown that Nobeling spaces are uniquely determined by the universal extension and embedding properties.