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 XnJ and ildeXnJ. result Explicit calculations of inverse metric matrices for n=2. New geometry based on Siegel upper half-space with volume formula.
problem Developing a new 3D geometry based on Siegel upper half-space.
method Constructing a geometry fibered over Siegel upper half-space and providing a volume formula.
result Volume of Siegel-Seifert closed manifolds is the fiber circle length times base manifold's Euler characteristic.
Extended metric defined on Siegel-Jacobi space using invariant forms.
problem Defining a metric on the extended Siegel-Jacobi upper half space.
method Matrix embedding, pre-Iwasawa decomposition, invariant forms, sum of squares of forms.
result Invariant metric on the extended Siegel-Jacobi upper half space is derived.
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
Researchers explore geometric dualities in statistical manifolds.
problem Understanding geometric dualities in statistical manifolds.
method Exploring the dualistic geometry of statistical manifolds, focusing on Hessian manifolds.
result Moduli space of univariate normal distributions corresponds to Siegel half-space and Siegel-Jacobi space.
In this paper, we first single out a proper subgroup Γof Sp(4,Z) generated by three elements, which arises from the parallelogram decompositions of translation surfaces in H(2). We then prove that the space H(2)/C* can be identified to the quotient J_2/Γ, where J_2 is the Jacobian locus in the Siegel upper half space H…
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.
We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomor…
Geodesics on extended Siegel-Jacobi upper half-plane determined.
problem Determining geodesics on a complex geometric space.
method Equating parameters in geodesic equations on the extended Siegel-Jacobi upper half-plane.
result Geodesic equations on Siegel-Jacobi, Siegel, and Heisenberg spaces.
Estimates Bergman kernel for Siegel varieties, focusing on geodesic distances.
problem Estimating the Bergman kernel for complex manifolds of Siegel varieties.
method Analyzes geodesic distances and derives estimates for the Bergman kernel.
result Derives estimates of the Bergman kernel for Siegel varieties.
Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.
problem Characterizing the geometry of the extended Siegel-Jacobi upper half-plane.
method Parameterized using S-coordinates and expressed in terms of invariant metrics.
result Extended Siegel-Jacobi upper half-plane is a reductive, non-symmetric manifold.
The Bergman kernel and period map for curves are studied.
problem Characterize the Torelli map's second fundamental form on the moduli space of curves.
method Use the Bergman kernel form associated to the curve and its harmonic representative.
result The Bergman kernel form is the harmonic representative of the multiplication by a certain meromorphic form.
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…
We show that for any weakly convergent sequence of ergodic SL2(R)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-M…
We find the homogenous Kähler isomorphism FC which expresses the Kähler two-form on the Siegel-Jacobi domain D1J=C×D1 as the sum of the Kähler two-form on C and the one on the Siegel ball D1. The classical motion and quantum evolution on D1J…
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.
We provide a generalization of the Deligne sheaf construction of intersection homology theory, and a corresponding generalization of Poincaré duality on pseudomanifolds, such that the Goresky-MacPherson, Goresky-Siegel, and Cappell-Shaneson duality theorems all arise as special cases. Unlike classical intersection homo…
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.
Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the 2n−1-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincide…
Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
The paper extends Siegel-Veech formula to convex flat cone spheres.
problem No formula exists for flat surfaces with irrational cone angles.
method Defined a generalized Siegel-Veech transform and Siegel-Veech measure.
result The Siegel-Veech measure is absolutely continuous and piecewise real analytic.
We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.
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 …
Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of Rn by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented t…
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.
We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in Cn. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk D1J, which is a parti…
Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.
problem Linking numbers between geodesics in arithmetic hyperbolic 3-folds.
method Analyzing integral of Kudla--Millson theta series over Seifert surfaces.
result Series converges to genus 2 Siegel modular forms of weight 2.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
Two Kähler structures are PCR equivalent in the Siegel domain.
problem Equivalence of two Kähler structures in the Siegel domain.
method Construction of complex hyperbolic and Kähler structures from Sasakian structure.
result PCR Kähler equivalent structures in Siegel domain.
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.
We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.
Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. We show how to find each such polyhedron in the upper half-space model by considering lines and circles in the plane. Finally, we give matrix generators in $\mathrm{…
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
Developed a half-space model for pseudo-hyperbolic space.
problem Modeling pseudo-hyperbolic space for any dimensions.
method Created an isometric embedding of pseudo-hyperbolic space into a half-space.
result Geodesics, totally geodesic submanifolds, horospheres, and isometry group are described in the half-space model.
Paper introduces capillary Schwarz symmetrization in half-space.
problem Capillary problems in half-space.
method Introduces a special anisotropic gauge to transform capillary symmetrization to convex symmetrization.
result Capillary Schwarz symmetrization in half-space established.
Computes constants for specific geometric structures.
problem Calculating constants for specific geometric structures.
method Analyzes saddle connections and Prym eigenforms.
result Computed Siegel-Veech constants for real quadratic orders.
The paper establishes inequalities for p-capacitary functions in flat half-spaces.
problem Understanding p-capacitary functions in asymptotically flat half-spaces. method Establishes monotone quantities and mass-capacity inequalities.
result Sharp inequalities attain equality on a Schwarzschild half-space.
We relate trimmed sums of twists in cylinders along a typical Teichmuller geodesic to the area Siegel-Veech constant.
Study proves no nontrivial minimal surfaces in half-space with specific boundary conditions.
problem Proving the nonexistence of minimal surfaces in half-space with certain boundary conditions.
method Analyzes minimal surface equations in half-space with specific boundary conditions.
result Establishes Liouville type theorems for minimal surfaces in half-space.
The real Jacobi group G1J(R), defined as the semi-direct product of the group SL(2,R) with the Heisenberg group H1, is embedded in a 4×4 matrix realisation of the group Sp(2,R). The left-invariant one-forms on G1J(R) and their dual orthogonal left-i…
Proposes Siegel neural networks for improved classification tasks.
problem Classification on Siegel spaces is underexplored.
method Uses quotient structure and vector-valued distance notation.
result Demonstrates state-of-the-art performance in radar and node classification.
Genetic Algorithm improves Nelson-Siegel-Svensson model calibration for interest rates.
problem Calibrating the Nelson-Siegel-Svensson model is difficult due to nonlinearity and parameter co-dependence.
method Applied Genetic Algorithm to optimize model parameters.
result Constructs stable interest rate curves and model parameters over time.
The study predicts large genus behavior of quadratic differential volumes and constants.
problem Predicting large genus behavior of quadratic differential volumes and constants.
method Analyzing conjectures on asymptotic behavior of Masur-Veech volumes and area Siegel-Veech constants.
result Conjectures on large genus asymptotics of quadratic differential volumes and constants.
Study orbits of discrete lattice actions on the plane, derive new results for Veech surfaces.
problem Count pairs of holonomy vectors in Veech surfaces with bounded parameters.
method Siegel-Veech-type integral formula for averages of pairs of orbits.
result Upper bounds on pairs of holonomy vectors in Veech surfaces with bounded parameters.