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…
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48 results for “Jacobi algorithm”
Paper solves investment strategy optimization with deep learning.
problem Maximizing investor utility with optimal asset allocation.
method Solves PDEs with Deep Galerkin method.
result Deep learning algorithm outperforms finite difference method.
New method finds vector fields with maximal Jacobi operator rank in manifolds.
problem Existence of non-isotropic vector fields with maximal rank Jacobi operator.
method Effective algorithmic procedure using a quadratic parallel differential form.
result Existence of non-isotropic vector field with maximal rank Jacobi operator implies manifold is locally non-reducible.
Deep learning method proves convergence for solving HJI equations.
problem Solving Hamilton-Jacobi-Isaacs equations for reachability analysis.
method Uniform convergence guarantee for DeepReach algorithm.
result DeepReach algorithm converges to classical solution of HJI equation.
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 solve continuous-time reinforcement learning using distributional Hamilton-Jacobi-Bellman equations.
problem Predicting the distribution of returns in continuous-time, stochastic environments.
method We derive a distributional Hamilton-Jacobi-Bellman equation for Itô diffusions and Feller-Dynkin processes, and propose an algorithm based on a JKO scheme.
result We propose an online control algorithm that can be used to approximately solve the distributional HJB equation.
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.
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.
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…
Paper solves complex stochastic control problems with a new algorithm.
problem Non-Markovian stochastic optimal control with semilinear SHJB equations.
method Policy-iteration algorithm based on successive linearization.
result Approximation sequence converges monotonically to the value function with exponential rate.
Study on Tukey depth in machine learning using Hamilton-Jacobi equations.
problem Understanding Tukey depth in machine learning applications.
method Derive necessary conditions for Tukey depth in continuum limit, formulating them as a Hamilton-Jacobi equation.
result Prove existence and uniqueness of viscosity solutions for the derived equation, which bounds Tukey depth.
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…
Holomorphic Jacobi Manifoldsmath.DG
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi str…
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.
The paper proves normal forms for Dirac-Jacobi bundles and splitting theorems for Jacobi structures.
problem Proving normal forms and splitting theorems for Jacobi structures.
method Using recent techniques from Bursztyn, Lima and Meinrenken, the paper proves normal forms for Dirac-Jacobi bundles and splitting theorems for Jacobi pairs.
result The paper provides an alternative proof of the splitting theorem of homogeneous Poisson structures.
Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.
problem Calculating inverse metric matrices on Siegel-Jacobi spaces.
method Inversion of metric matrices on and .
result Explicit calculations of inverse metric matrices for .
We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial …
Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the …
Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds are reviewed. We present both in a unified approach using the representation of the Lie algebra of functions on itself by means of the hamiltonian vector fields. The use of the associated Lie algebroid allows to prove that the Lichnerowicz-Jacobi cohomolog…
Paper introduces stochastic HJB on Jacobi structures.
problem Stochastic analysis on Jacobi manifolds.
method Global stochastic analysis techniques, extending Bismut and Lázaro-Camí work.
result Proposes a stochastic HJB framework.
Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids.
Jacobi Structures in math-ph
The most general Jacobi brackets in are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
Berry phases and connection matrices defined on homogeneous spaces attached to Siegel-Jacobi groupsmath.DG
Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.
problem Understanding Berry phases and connection matrices on Siegel-Jacobi spaces.
method Examines the Siegel-Jacobi disk and upper half-plane, calculates connection matrices and covariant derivatives.
result Calculates the connection matrix and covariant derivatives on the extended Siegel-Jacobi upper half-plane.
New algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an algebra is associated with each Dirac-Jacobi structure.
result There is a one-to-one correspondence between MC elements of the algebra and small deformations of the Dirac-Jacobi structure.
New characterization of Osserman tensors using Jacobi-orthogonality.
problem Characterizing Osserman tensors.
method Introducing Jacobi-orthogonality as a new potential characterization.
result Jacobi-orthogonal tensors are Osserman, and all known Osserman tensors are Jacobi-orthogonal.
A new definition for vector fields extends the Jacobi set concept.
problem Describing interactions between vector fields on complex domains.
method Piecewise linear approach for simplicial complexes.
result Generalizes Jacobi set concept to vector fields.
Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.
problem Analyzing Hamilton-Jacobi theory across different geometric backgrounds.
method Geometric review of Hamilton-Jacobi theory, focusing on Jacobi and Leibniz identities.
result Novel Hamilton-Jacobi equation for conformal Hamiltonian vector fields.
New action functionals for sigma models on Jacobi bundles.
problem Defining action functionals for sigma models on Jacobi bundles.
method Generalizing Jacobi brackets to Jacobi bundles and proposing different approaches.
result Solutions correspond to morphisms of Jacobi algebroids.
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.
Improved gossip algorithm for networks of given dimension using Jacobi polynomial iterations.
problem Efficiently estimating the average of values in a network of agents.
method A gossip algorithm that depends only on the spectral dimension of the network, using Jacobi polynomial iterations.
result Significantly faster convergence compared to previous methods in the non-asymptotic regime.
The paper outlines key geometry issues in Siegel-Jacobi space.
problem Basic problems in the geometry of the Siegel-Jacobi space.
method Proposes basic problems.
result Outlines key geometry issues in Siegel-Jacobi space.
Abstract: From complex financial models to simpler Hamilton-Jacobi equations.
problem Complex financial models for multi-dimensional Black-Scholes.
method Linked Hamilton-Jacobi equations to simplify financial models.
result Simplified financial models using Hamilton-Jacobi equations.
Defines Jacobi fields in nonholonomic mechanics.
problem No specific problem stated; focuses on definition and properties.
method Characterizes Jacobi fields using Riemannian geometry methods and finds them explicitly. Uses complete lift and curvature/torsion of nonholonomic connection.
result Derives nonholonomic Jacobi equations in two equivalent ways.
We analyze the relationship between the covering of the Jacobi group and the squeezed states. We attach some nonclassical states to the Jacobi group. The matrix elements of the Jacobi group are presented.
The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.
problem Classifying real hypersurfaces with a particular Jacobi operator.
method Introducing and classifying real hypersurfaces with a quadratic Killing structure Jacobi operator.
result A classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator.
Equations of motion for linear Hamiltonians in the real Jacobi group
problem Equations of motion for linear Hamiltonians in the real Jacobi group
method Using the energy function on the extended Siegel-Jacobi upper half space
result Equations of motion attached to linear Hamiltonians in the generators of the real Jacobi group
Defines gauge transformations for Jacobi structures and their effects on contact groupoids.
problem Understanding transformations of Jacobi structures and their implications.
method Definition and discussion of gauge transformations for Jacobi structures and their impact on contact groupoids.
result Gauge transformations affect the contact structure of contact groupoids.
Holomorphic Jacobi manifolds integrate to complex contact groupoids.
problem Integrating holomorphic Jacobi manifolds.
method Homogenization scheme to identify and integrate holomorphic Jacobi manifolds to complex contact groupoids.
result Holomorphic Jacobi manifolds integrate to complex contact groupoids.
In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive Jacobi-like identities of tangent-vector-valued forms from the general Jacobi identity.
In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown that Jacobi algebroids can be understood in terms of a kind of curved Q-manifold, w…
Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operatormath.DG
No real hypersurfaces found in complex Grassmannians with specific Jacobi operators.
problem Existence of real hypersurfaces in complex Grassmannians with semi-parallel structure Jacobi operator.
method Proved nonexistence through mathematical analysis.
result No real hypersurfaces exist in complex Grassmannians of rank two with semi-parallel structure Jacobi operator.
Study contact resolutions for Jacobi structures, providing examples and impossibility results.
problem Understanding contact resolutions of Jacobi structures.
method Examining various classes of Jacobi structures and their contact properties.
result Identified conditions under which contact resolutions exist and those where they do not.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding compatibility between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving it for specific structures.
result Fundamental examples of Jacobi structures lead to known structures like Riemann-Poisson, Kenmotsu, and locally conformally Kähler.
Unified Jacobi coupling construction for various geometric settings.
problem Constructing Jacobi structures on associated bundles.
method Extending Sternberg--Weinstein coupling to Jacobi geometry.
result Associated bundles inherit Jacobi structures from base spaces.
Develops integrators for Hamiltonian systems in Jacobi manifolds.
problem Modeling conservative systems with dissipative and thermodynamic phenomena.
method Constructs structure-preserving integrators for Hamiltonian systems in Jacobi manifolds.
result Proposes a numerical integration technique compatible with Jacobi dynamics.