The paper proves stability for contact groupoids and deformations.
problem Stability of contact groupoids and deformations.
method Proof of Gray stability for compact contact groupoids.
result Stability results for deformations of induced Jacobi bundles.
Study on stability of 3D sessile drops, identifying degenerate kernel.
problem Linear stability of three-dimensional sessile drops with a free contact line.
method Derived constrained second variation, formulated Jacobi problem, combined geometric and Fourier analysis.
result Kernel of the constrained Jacobi operator is exactly the space of horizontal translations under pressure-volume nondegeneracy.
The study refines stability results for Yang-Mills fields and harmonic maps.
problem Stability of Yang-Mills fields and harmonic maps.
method Refinement of stability results using Jacobi operator over S^m.
result Refined stability results and Morse index estimates.
A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jac…
In this article, we investigate the stability of leaves of minimal foliations of arbitrary codimension. We also study relations between Jacobi fields and vector fields which preserves a foliation and we use these results to Killing fields.
Investigates harmonic self-maps' stability on cohomogeneity one manifolds.
problem Stability of harmonic self-maps on cohomogeneity one manifolds.
method Systematic study of Jacobi equation for harmonic self-maps.
result Explicit solutions for specific cases show identity map's stability.
Develops multivalued Jacobi fields for stable submanifolds.
problem Stability of minimally immersed submanifolds in Riemannian manifolds.
method Defines and studies multiple valued Jacobi fields as minimizers of a second variation functional.
result Any Q-valued Jacobi field can be decomposed into classical Jacobi fields except on a singular set of codimension at least two.
Develops KCC theory for higher-order dynamical systems.
problem Investigates properties of higher-order dynamical systems.
method Introduces a geometric description of dynamical systems using Finsler spaces and non-linear connections.
result Only even-dimensional dynamical systems can exhibit both Jacobi stability and instability, while odd-dimensional systems are always Jacobi unstable.
Stability inequalities for specific solutions in high dimensions.
problem Stability of solutions to the one-phase Bernoulli problem.
method Proving strict stability inequalities for cohomogeneity one solutions with bi-orthogonal symmetry.
result Strict stability for cohomogeneity one solutions in dimensions 7 and above.
The study examines stability and classification of special minimal hypersurfaces in high dimensions.
problem Stability and classification of special minimal hypersurfaces in high dimensions.
method Analysis of stability and nondegeneracy properties using Jacobi fields and Morse index.
result In high dimensions, these hypersurfaces are strictly stable and have a full classification of bounded Jacobi fields.
Study stability of multiphase partitions with volume constraints.
problem Stability of multiphase partitions in convex domains.
method Detailed derivation of second variation formula and study of Jacobi operator eigenvalues.
result Derive stability criteria, including recapturing Sternberg-Zumbrun instability result.
New perspective on KCC theory for dynamical systems.
problem Geometric description of dynamical systems.
method Introducing a non-linear and Berwald type connection to describe dynamical systems geometrically.
result Established the relationship between linear and Jacobi stability for two-dimensional autonomous systems.
Develops Lagrange-Hamilton geometry for COVID-19 disease dynamics.
problem Modeling the spread of COVID-19 disease.
method Least squares variational method, nonlinear connections, d-torsions, Lagrangian Yang-Mills.
result Jacobi stability of the dynamical system.
Market makers optimize trading with a new implicit scheme for complex inequalities.
problem Optimizing trading in a limit order book with stochastic and impulse control.
method Implicit numerical scheme coupled with policy iteration algorithm.
result Convergence to the unique viscosity solution of the HJBQVI.
Unified approach to equity markets with open and hybrid Jacobi models.
problem Stochastic Portfolio Theory problems in equity markets.
method Combining open markets and hybrid Jacobi processes.
result Stability of capital distribution curve and growth optimal strategies.
Study on stability in discretized hydrodynamics model.
problem Stability analysis of discretized hydrodynamics model.
method Geometric structure of Euler equations, convergence of sectional curvature and Jacobi equations.
result Geometric insights from discretized model transfer to Euler equations.
The paper examines the stability of Killing cylinders in hyperbolic space.
problem Stability of Killing cylinders in hyperbolic space.
method Explicit computation of Morse index for Jacobi operator on various support surfaces.
result Delaunay surfaces can be bifurcated from Killing cylinders supported on geodesic planes.
In this paper, we study submanifolds with constant rth mean curvature Sr. We investigate, the stability of such submanifolds in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros - Sousa and Alías - Colares, concerning conformal fields, to an arbitrary manifol…
Compactness of minimal hypersurfaces with volume and eigenvalue bounds.
problem Proving compactness of minimal hypersurfaces with volume and eigenvalue constraints.
method Analyzing minimal hypersurfaces on closed Riemannian manifolds with specific eigenvalue and volume conditions.
result Strong convergence to a smooth limit submanifold away from at most p-1 points when p-th Jacobi eigenvalue is bounded.
In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which are explicitly constructed and shown to give a complete description of the enti…
The paper computes spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.
problem Computing spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.
method Analyzing eigenvalues of second order Hill's equations and proving inequalities for stability index and eigenvalues.
result Proves that the stability index of minimal rotational examples is greater than 3n+4 and there are at least 2 positive Laplacian eigenvalues smaller than n. Study moduli space of quadratic differentials with new geometric insights.
problem Understanding the structure of moduli spaces of quadratic differentials.
method Using decorated marked surfaces, Abel-Jacobi map, and 3-Calabi-Yau categories.
result Fundamental group of moduli space equals kernel of Abel-Jacobi map.
Stability of Morse index for harmonic maps on degenerating surfaces analyzed.
problem Analyzing stability of Morse index for harmonic maps on degenerating Riemann surfaces.
method Analysis of second variation of energy, identification of conditions for upper semicontinuity, explicit contribution of geodesics.
result Sharper control of spectrum of Jacobi operator, explicit contribution of geodesic segments to Morse index.
Optimizes dividends with stability for risky businesses.
problem Maximizing dividends with stability in risky businesses.
method Linear-quadratic optimization for a general Lévy process.
result Derives optimal affine dividend strategies with stability.
Study stability and bifurcation of liquid interfaces in cylindrical supports.
problem Stability and bifurcation of liquid interfaces in cylindrical support surfaces.
method Analysis of eigenvalues of the Jacobi operator, Plateau-Rayleigh instability, bifurcation theory.
result Conditions for the emergence of new morphologies and bifurcations from circular cylinders.
The Lawson surface ξ_{g,1} has index 2g+3 and nullity 6.
problem Characterizing the geometric properties of Lawson surfaces.
method Analyzing the linearized stability of the Lawson surface.
result The Lawson surface ξ_{g,1} has no exceptional Jacobi fields.
The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…
Global Morse index theorem applied to Jacobi fields on CMC surfaces.
problem Existence and structural theorem of Jacobi fields on CMC surfaces.
method Global Morse index theorem proof via set-continuity of domain shapes and eigenvalue continuity.
result Global Morse index theorem provides structural existence of Jacobi fields.
The paper sets a lower bound for the stability index of compact constant mean curvature surfaces.
problem Determining the stability index of compact constant mean curvature surfaces.
method Proving a linear lower bound for the stability index using geometric and spectral comparisons.
result The stability index is bounded below by a linear function of the genus.
Reconstructs piecewise constant functions from geodesic integrals.
problem Recovering piecewise constant functions from X-ray data.
method Injectivity proof using variations through geodesics, improved for simple manifolds.
result Explicit formulas for function values near the boundary and stability analysis.
Develops geometry for Lotka-Volterra model of species competition.
problem Population dynamics of competing species.
method Least squares variational method, Lagrange-Hamilton geometry.
result Jacobi stability discussed for the Lotka-Volterra system.
Bidirectional bounds stabilize training of energy-based models.
problem Training energy-based models is difficult and prone to instability.
method Propose bidirectional bounds linking to gradient penalty and Jacobi-determinant estimator.
result Significant stabilization and high-quality density estimation achieved.
Stability of Minkowski space proved for massless particles in Einstein-Vlasov system.
problem Global stability of Minkowski space for massless particles in the Einstein-Vlasov system.
method Proof by showing matter supported in the wave zone, semi-global existence for characteristic initial value problem, weighted estimates for Jacobi fields.
result Global stability of Minkowski space for massless particles in the Einstein-Vlasov system.
Let Σ be a compact immersed surface with constant weighted mean curvature Hf in a weighted manifold (M3,g,f). In this paper we obtain upper bounds for the first eigenvalue of the weighted Jacobi operator on Σ in terms of Hf and the curvature of the ambient. As consequence we obtain that there is no stable …
We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce t…
The Runge-Kutta-Legendre scheme improves pricing American options and other derivatives.
problem Pricing American options and other derivatives with improved accuracy and stability.
method Runge-Kutta-Legendre finite difference scheme applied to Black-Scholes and Heston models.
result Improved convergence and stability compared to existing schemes.
We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providin…
We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to co…
The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.
problem The behavior of singular minimal hypersurfaces in dimensions 7 and above.
method Analyzes the local behavior of minimal hypersurfaces under perturbations and convergence of families of hypersurfaces.
result Existence and smoothness of nearby minimal hypersurfaces under perturbations, uniqueness of homological minimization, and existence of Jacobi fields.
Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
problem No specific problem stated; focuses on new structure definition.
method Definition and properties of Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
result Defines a new structure on Jacobi-left-symmetric algebroids.
Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
Modeling financial systemic risk with optimal control theory for stability.
problem Analyzing and stabilizing systemic risk in interconnected financial entities.
method Developed a theoretical model using optimal control theory, including steps for synthesizing stabilizing controllers.
result The model ensures that the H∞ norms of the mappings from disturbance to output are less than a predefined constant, stabilizing the system. We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissoniza…
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
problem Generalizing Dirac pairs to Jacobi algebroids.
method Introducing Dirac pairs on Jacobi algebroids and showing their relationship to Lie algebroids.
result Dirac pairs on Jacobi algebroids characterize compatible structures.
We use the supergeometric formalism, more precisely, the so-called "big bracket" (for which brackets and anchors are encoded by functions on some graded symplectic manifold) to address the theory of Jacobi algebroids and bialgebroids (following mainly Iglesias-Marrero and Grabowski-Marmo as a guideline). This formalism…
Defines compatibility between Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
problem Generalizing compatibility between Poisson and pseudo-Riemannian metrics to Jacobi structures.
method Introduces and studies compatibility conditions for Jacobi structures and pseudo-Riemannian metrics on Jacobi algebroids.
result Compatibility conditions are preserved under Poissonization and equivalent to Sasakian structures for contact pseudo-metrics.
This paper extends Jacobi field theory to Jacobi curves and their curvatures.
problem Characterizing and understanding Jacobi curves and their curvatures.
method Developed a new theory of Jacobi curves and associated curvatures, derived Ricci curvature, and presented a Cartan-like theory.
result Jacobi curves are fully characterized by a family of conformal symplectic invariant curvatures.