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 …
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48 results for “Jacobi polynomial”
Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
problem Understanding polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
method Analyzing polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.
result Results generalize previous work by Katada and study polynomiality and outer nature of these functors.
Study of zonal spherical functions on partial flag manifolds using Jacobi polynomials.
problem Understanding zonal spherical functions on partial flag manifolds.
method Matrix variate version of Koornwinder's method for constructing orthogonal polynomials, using Hermitian Jacobi polynomials and multivariate Schur polynomials.
result Conjecture that Hermitian Jacobi polynomials are elementary zonal spherical functions.
A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.
A representation of the Jacobi algebra by first order differential operators with polynomial coefficients on the manifold is presented. The Hilbert space of holomorphic functions on which the holomorphic first order differential operators with …
The study confirms a conjecture about polynomials related to symmetric spaces.
problem Understanding polynomials associated with isotropy orbits of symmetric spaces.
method Identified Reiswich's polynomials as special cases of Jacobi polynomials and proved their conjecture.
result The polynomials have pairwise different real roots in the interval [0,1].
Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introd…
Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We give an explicit and unified algebraic procedure for the construction of the ca…
The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.
problem Understanding the n-loop Kontsevich invariant for knots with identical Alexander polynomials.
method Analyzes the subspace generated by the n-loop Kontsevich invariant of knots with genus ≤ g and same Alexander polynomial.
result For n ≥ 2, the subspace is finite-dimensional.
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.
We present a holomorphic representation of the Jacobi algebra by first order differential operators with polynomial coefficients on the manifold . We construct the Hilbert space of holomorphic functions on which these differential operators a…
Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp bounds for eigenfunctions on products of rank-one symmetric spaces.
Quantization goes Polynomialq-fin.PR
Paper applies quantization to polynomial processes for option pricing.
problem Quantization for polynomial processes in finance.
method Two quantization procedures for stochastic volatility Jacobi process.
result Theoretical and practical tools for fast option pricing.
Study automorphism groups' action on Jacobi diagrams, leading to new decompositions.
problem Understanding automorphism groups' actions on Jacobi diagrams.
method Analyzing the induced actions on graded vector spaces and constructing polynomial functors.
result Indecomposable decomposition of and polynomial functor construction.
A fast numerical method for pricing double barrier options using Lagrange interpolation.
problem Pricing discrete double barrier knock-out call options efficiently.
method Approximating recursive solutions of the heat equation with Lagrange interpolation on Jacobi polynomials nodes.
result The method significantly reduces CPU time as the number of monitoring dates increases.
Geodesics in Jet Spacemath.OC
Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.
problem Characterize geodesics in jet space and identify those that are globally minimizing.
method Sub-Riemannian geometry, Hamilton-Jacobi equations, and analysis of period degenerations.
result Some polynomials yield globally minimizing geodesics, with conjectures on the independence of cut time.
In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jaco…
Polynomial expansions improve option pricing accuracy.
problem Efficiently pricing and Greeks in stochastic volatility models.
method Analytic series representations for European and exotic options.
result Polynomial expansions match Fourier transform accuracy.
New contact structures on spheres and tori with weakly compatible metrics.
problem Contact sphere theorem does not hold for weakly compatible metrics.
method Construction of nonvanishing curl eigenfields using Jacobi or trigonometric polynomials.
result Geometric rigidity for tight contact structures on the 3-sphere.
In this text we give a decomposition result on polynomial poly-vector fields generalizing a result on the decomposition of homogeneous Poisson structures. We discuss consequences of this decomposition result in particular for low dimensions and low degrees. We provide the tools to calculate simple cubic Poisson structu…
New method uses TT approximations to solve HJB equations for efficient sampling.
problem Efficiently sampling from complex probability densities.
method Direct time integration of HJB equations using Tensor Train compression.
result Sample-free, dimensionality-avoiding integration method.
Investigates polynomial solutions to minimal surface equation, proving constraints and structure theorems.
problem Finding polynomial solutions to the minimal surface equation.
method Proves structure theorem, analyzes polynomial constraints, and uses eigenvalue estimates.
result Polynomial solutions must contain terms of both high and low degree, and have specific factorization properties.
A new quantum relation connects exceptional Lie algebras and knots.
problem Understanding the relationship between exceptional Lie algebras and quantum invariants of knots.
method Developed a two-parameter skein relation on trivalent graphs that specializes to exceptional Lie algebras.
result Found a new quantum exceptional polynomial that agrees with classical computations for knots and links.
We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that case only a terminal condition is needed. Conversely, the case of contracts with …
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…
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…
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…
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 .
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.
Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.
problem Positivity of characteristic forms in vector bundles.
method Explicit computation of Chern curvature and use of flag bundles.
result Positivity of polynomials in Chern forms for Griffiths semipositive bundles.