Paper introduces economic Carnot cycles in Roegenian economics.
problem Economic efficiency limits in Roegenian systems.
method Uses Carnot cycle principles to model economic systems.
result Validates economic Carnot cycles in Roegenian economics.
We consider dynamics of financial markets as dynamics of expectations and discuss such a dynamics from the point of view of phenomenological thermodynamics. We describe a financial Carnot cycle and the financial analogue of a heat machine. We see, that while in physics a perpetuum mobile is absolutely impossible, in ec…
The paper analyzes liquidity in decentralized finance, deriving impact functions and de-pegging risks.
problem Understanding and quantifying market impact and de-pegging risk in decentralized finance.
method Derives market impact functions for optimal-growth liquidity providers, views Constant Product Market Maker as a Carnot engine, and links de-pegging risks to catastrophe bonds.
result New insights into liquidity models and de-pegging risks in decentralized finance.
This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (…
Smooth C1 contact maps are always smooth in rigid Carnot groups.
problem Smoothness of C1 contact maps in rigid Carnot groups. method Analyzing C∞-rigid Carnot groups to show C1-contact maps are smooth. result Smooth C1 contact maps are always smooth in rigid Carnot groups. Characterizes functions in Carnot groups of step 2.
problem Understanding intrinsic Lipschitz functions in Carnot groups.
method Characterization via intrinsic distributional gradients.
result Characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2.
Study of Sard problem in Carnot groups using dynamical systems.
problem Sard problem in sub-Riemannian Carnot groups.
method Dynamical-systems approach to study singular curves.
result Positively answer the Sard problem in some Carnot groups.
Improved Sobolev mappings in Carnot groups with weaker assumptions.
problem Improving Sobolev mappings in Carnot groups with weaker conditions.
method Using Buser-Karcher center-of-mass and polynomial expressions in moments.
result Rigidity and structural results hold under weaker Sobolev exponents.
Curves in Carnot groups avoid compact sets, growing at least t1/s.
problem Existence of periodic normal geodesics in subFinsler Carnot groups.
method Analysis of curves satisfying Pontryagin Maximum Principle.
result Normal curves in subFinsler Carnot groups leave every compact set.
New bounds on geodesic dimension and curvature exponent in Carnot groups.
problem Characterizing geodesic dimension and curvature exponent in Carnot groups.
method Characterization and lower bound calculation for geodesic dimension and curvature exponent.
result Found an example where curvature exponent is greater than geodesic dimension.
We prove that H-type Carnot groups of rank k and dimension n satisfy the MCP(K,N) if and only if K≤0 and N≥k+3(n−k). The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in…
Carnot groups can be polarized if they have specific coordinate systems.
problem Understanding when Carnot groups can be polarized.
method Proving Carnot groups with certain coordinate systems are polarizable.
result Carnot groups with suitable horizontal polar coordinates are polarizable.
Compact currents and charges in Carnot groups proved.
problem Compactness of normal currents in Carnot groups.
method Dual compactness argument for Rumin forms using pseudo-differential calculus.
result Compactness of normal currents in Carnot groups in flat topology.
ODE trajectories become abnormal curves in Carnot groups.
problem Understanding abnormal curves in Carnot groups.
method Explicit construction of covectors for abnormal curves.
result Polynomial ODE trajectories lift to abnormal curves in Carnot groups.
Study on mappings between nonrigid Carnot groups, proving quasisymmetric rigidity.
problem Quasisymmetric homeomorphisms in nonrigid Carnot groups.
method Use pullback theorem from previous work to show reducibility and rigidity.
result Quasisymmetric homeomorphisms are reducible in nonrigid Carnot groups, except for specific cases.
Study intrinsic regular surfaces in Carnot groups, generalizing results from Heisenberg groups.
problem Equivalence of definitions of intrinsic regular surfaces in Carnot groups.
method Generalize results from Heisenberg groups to Carnot groups.
result Equivalence of definitions of intrinsic regular surfaces in Carnot groups.
New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
This paper studies rectifiability in Carnot groups and proves geometric area formulas.
problem The study of rectifiability in Carnot groups and related geometric properties.
method Analysis of rectifiable measures in Carnot groups, geometric area formulas, and rectifiability of geodesic spheres.
result Geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in Carnot groups.
Commutes Pansu pullback with spectral complexes in Carnot groups.
problem Understanding the relationship between Pansu pullback and spectral complexes in Carnot groups.
method Proving commutativity between Pansu pullback and differentials in spectral complexes.
result Commutes Pansu pullback with spectral complexes in Carnot groups.
Study on mappings in Carnot groups, proving rigidity results.
problem Understanding mappings in Carnot groups and proving rigidity.
method Structural results for Sobolev mappings, proving rigidity or regularity.
result Establishes partial rigidity and partial regularity theorems.
The paper explores the Rumin complex and spectral sequence on Carnot groups.
problem Understanding the relationship between Rumin complex and spectral sequence on Carnot groups.
method Investigates the Rumin complex and spectral sequence on Carnot groups, focusing on the filtration by homogeneous weights.
result Provides a detailed insight into the relationship between the Rumin complex and the spectral sequence on Carnot groups.
Study shows a specific Carnot group violates a curvature exponent bound.
problem Understanding the curvature exponent in step-two Carnot groups.
method Examined convergence of Lie algebra structure constants.
result Found a Carnot group where curvature exponent bound is violated.
Maps commuting with sub-Laplacians on Carnot groups are conformal.
problem Characterizing maps preserving sub-Laplacians on sub-Riemannian Lie groups.
method Analyzing smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians.
result Sub-Laplacian determines the sub-Riemannian structure in Carnot groups.
Maps in Carnot groups are equivalent to solutions of a PDE system.
problem Understanding maps in Carnot groups of step 2.
method Equivalence between intrinsic Lipschitz maps and solutions to a PDE system.
result Intrinsic Lipschitz maps are equivalent to weak solutions of a PDE system.
New insights into integrability and rectifiability in sub-Riemannian geometry.
problem Understanding rectifiability in sub-Riemannian spaces.
method Refined Frobenius Theorem for non-involutive distributions, new metric space class.
result Carnot-Carathéodory spaces are extremal in rectifiability.
We characterize the rigidity of Carnot groups in the class of C2 contact maps in terms of complex characteristics. Furthermore, we obtain a Liouville type theorem for Carnot groups which states that 1-quasiconformal maps form finite dimensional Lie groups.
Existence and rigidity results for lifts in Carnot groups.
problem Existence and properties of lifts for maps between Carnot groups.
method Use central extensions to define lifts and prove existence and rigidity results for Lipschitz, Sobolev, and quasiconformal maps.
result Quasiconformal maps admit contact lifts that are bi-Lipschitz.
Extends potential theory to Carnot groups, estimating Hausdorff dimension.
problem Estimating Hausdorff dimension of polar sets in Carnot groups.
method Geometric completeness and Riesz potential inequalities in Carnot groups.
result Developed applications in CR geometry and quaternionic CR geometry.
We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.
Paper proves equivalence of derivatives for maps between Carnot groups.
problem Maps between Carnot groups and their derivatives.
method Elementary proof using Euclidean arguments and mean value estimates.
result Maps preserving horizontal curves are continuously Pansu differentiable.
Generalizes Cheeger inequality to Carnot-Carathéodory spaces.
problem Lower bounds on eigenvalues of Laplacians in complex spaces.
method Geometric approach, including Neumann and mixed boundary conditions.
result Concrete method to lower bound Cheeger constant.
Optimal quantization of measures on Carnot groups
problem Quantization of probability measures on Carnot groups
method Zador-type asymptotic formula and weak convergence of empirical measures
result Convergence of quantization error and density of absolutely continuous part
Study uniformly differentiable graphs in Carnot groups, proving area formulas.
problem Characterize uniformly differentiable intrinsic graphs in Carnot groups.
method Characterize uniform intrinsic differentiability via Hölder properties of projections of vector fields.
result Explicit area formula for uniformly intrinsically differentiable maps in Carnot groups.
We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.
This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.
problem Characterizing semigenerated Carnot groups and their applications to rectifiability.
method Algebraic approach focusing on semigroup generation and Engel-type quotients.
result Complete characterization of semigeneration in Carnot groups of step 3 and sufficient criteria for semigeneration in Carnot groups of arbitrary step.
We develope basic geometric quantities and properties of hypersurfaces in Carnot groups.
Study spectral properties of sub-Laplacians in Carnot groups.
problem Spectral properties of sub-Laplacians in Carnot groups.
method Proved pure point spectrum and spectral gap; applied to small ball problem and heat content.
result Proved existence of spectral gap and pure point spectrum.
A Carnot group G admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve γ in G and ε>0, there is a C1 horizontal curve Γ such that Γ=γ and Γ′=γ′ outside a set of measure at most ε. We verify this property for free Carno…
The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.
problem Exploring non-smooth geometries on Lie groups and their applications.
method Study of left-invariant metrics on Lie groups, focusing on nilpotent and Carnot groups.
result Illustrates the role of metric Lie groups, particularly Carnot groups, in various mathematical contexts.
Study Sard problem in step 2 and filiform Carnot groups.
problem Understanding the Sard problem in specific types of Carnot groups.
method Analyzing endpoint maps in step 2 and filiform Carnot groups.
result Characterized abnormal set in filiform groups and provided bounds in step 2 Carnot groups.
In Carnot groups, directional pliability allows curve extensions and approximations.
problem Existence of curve extensions and approximations in Carnot groups.
method Directional pliability in subsets of directions guarantees Whitney-type extensions and Lusin approximations.
result Every horizontal curve in the Engel group intersects a C1 curve in a set of positive measure. Study area and coarea formulas for graphs and submanifolds in Carnot groups.
problem Understanding geometric properties of submanifolds in Carnot groups.
method Developed area and coarea formulas for CH1 intrinsic graphs and submanifolds. result Deduced density properties for Hausdorff measures and coarea formula for Carnot groups.
Rectifiability shown for sub-Riemannian manifolds with Carnot tangent structure.
problem Understanding rectifiability in sub-Riemannian manifolds with specific tangent properties.
method Analyzing nilpotentization and embedding properties of sub-Riemannian manifolds into Carnot groups.
result Sub-Riemannian manifolds are countably rectifiable under certain conditions.
We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-…
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remar…
Study properties of sets with constant normal in Carnot groups.
problem Properties of sets with constant normal in Carnot groups.
method Analysis of subsets with intrinsic constant normal, proving regularity and structural results.
result Every constant-normal set in Carnot groups of step 4 or less is intrinsically rectifiable.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.