Finite Dimensional Approximation of the Monodromy Operator of a Periodic Delay Differential Equation with Piecewise Constant Orthonormal Functions

Using piecewise constant orthonormal functions, an approximation of the monodromy operator of a Linear Periodic Delay Differential Equation (PDDE) is obtained by approximating the integral equation corresponding to the PDDE as a linear operator over the space of initial conditions. This approximation allows us to consider the state space as finite dimensional resulting in a finite matrix approximation whose spectrum converges to the spectrum of the monodromy operator.

Singularities of regular black holes and the monodromy method for asymptotic quasinormal modes

We use the monodromy method to investigate the asymptotic quasinormal modes of regular black holes based on the explicit Stokes portraits.We find that,for regular black holes with spherical symmetry and a single shape function,the analytical forms of the asymptotic frequency spectrum are not universal and do not depend on the multipole number but on the presence of complex singularities and the trajectory of asymptotic solutions along the Stokes lines.

On Monodromy of Semi-stable Fibres

The fibration is one of the fundamental methods in the study of algebraic surfaces. In the early years, fibration was studied by using the method of complete classification of singular fibres, which was obtained through studying the combinatorical properties of singular fibres. But with the rising of the genus of fibration, this method will not work now. In 1977, a more essential classification of singular fibres of genus two was given by Eiji Horikawa by using relative canonical maps. Prof. Xiao Gang has successfully improved the classification of Horikawa and effectively studied some algebra.ic surfaces by using his classification. We know that, ordinarily, through the classification of singular fibres, parameters of fibration can be obtained. Based on this, we study the properties of surfaces. But for some parameters, we can effectively describe them only by the study of combinatorical properties and topological

A NEW PROOF OF THE HADAMARD MONODROMY THEOREM AND THE PERIODIC SOLUTION FOR A CLASS OF O.D.E.SYSTEMS AT RESONANCE

Early in 1904,Hadamard[1]showed the scale implicit function theorem of finitedimensions.Then it was generalized to infinite dimensions.In this paper a new proof of this theoremis given and the unique existence of the periodic solution for a class of ordinary differential equationsystems at resonance is obtained.

Two Classes of Solvable Subgroups of SL（3, C） and Application for Differential Equations in C

在 SL 的可解决的亚群的二个班的结构(3， C ) 在是在它的 monodromy 组是可解决的意义的 integrable 的 3 顺序 Fuchsian 方程的这份报纸，和 integrability 被给被讨论。

Evaluating accuracy of Hessian-based predictor-corrector integrators

Direct dynamics simulations are a useful and general approach for studying the atomistic properties of complex chemical systems because they do not require fitting an analytic potential energy function.Hessian-based predictor-corrector integrators are a widely used approach for calculating the trajectories of moving atoms in direct dynamics simulations.We employ a monodromy matrix to propose a tool for evaluating the accuracy of integrators in the trajectory calculation.We choose a general velocity Verlet as a different object.We also simulate molecular with hydrogen(CO_2) and molecular with hydrogen(H_2O) motions.Comparing the eigenvalues of monodromy matrix,many simulations show that Hessian-based predictor-corrector integrators perform well for Hessian updates and non-Hessian updates.Hessian-based predictor-corrector integrator with Hessian update has a strong performance in the H_2O simulations.Hessian-based predictor-corrector integrator with Hessian update has a strong performance when the integrating step of the velocity Verlet approach is tripled for the predicting step.In the CO_2 simulations,a strong performance occurs when the integrating step is a multiple of five.

Classical integrability of strings in γ-deformed backgrounds

This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied （TsT） transformations on S^3 and AdS3, respectively. It finds that the U（1） currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds generated by TsT transformations on both S3 and ADS3. Applying the TsT transformations, it derives the local Lax connections and the monodromy matrices in γ-deformed backgrounds with the spectral parameter which ensure the classical integrability of the string theories.

Asymptotic quasinormal modes of scalar field in a gravity's rainbow

In the framework of the gravity＇s rainbow, the asymptotic quasinormal modes of the modified Schwarzschild black holes undergoing a scalar perturbation are investigated. By using the monodromy method, we analytically calculated the asymptotic quasinormal frequencies, which depend on not only the mass parameter of the black hole, but also the particle＇s energy of the perturbation field. Meanwhile, the real parts of the asymptotic quasinormal modes can be expressed as TH In 3, which is consistent with Hod＇s conjecture. In addition, for the quantum corrected black hole, the area spacing is independent of the particle＇s energy, even though the area itself depends on the particle＇s energy. And that, by relating the area spectrum to loop quantum gravity, the Barbero-Immirzi parameter is given and it remains the same as from the usual black hole.

Arnold Tongues for Discrete Hill’s Equation

In this work we study two types of Discrete Hill’s equation. The first comes from the discretization process of a Continuous-time Hill’s equation, we called Discretized Hill’s equation. The Second is a naturally obtained in Discrete-Time and will be called Discrete-time Hill’s equation. The objective of discretization is preserving the continuous-time behavior and we show this property. On the contrary a completely different dynamic property was found for the Discrete-Time Hill’s equation. At the end of the paper is shown that both types share the nonoscillatory behavior of solutions in the 0-th Arnold Tongue.

Structure of solvable subgroup of SL(2,C) and integrability of Fuchsian equations on torus T2

The possible derived lengths and structures of solvable subgroups of SL(2,C) are given in Theorem 1,and the structures of the Riemann surfaces of solutions of the Fuchsian equation on torus with a solvable monodromy group are discussed.Examples in verifying the solvability of monodromy groups for some Fuchsian equations on T2 are given.

Integrability of second-order Fuchsian equation on Torus T^2

Consider the differential equationii-λp（t）u=0, （1）where the parameter γ∈R+, and t∈C u（t）∈C, and ρ（t） is the elliptic function ofWeierstrass with periods ω1,=2α and ω2=2αi （α∈R）. It is shown that ρ（t） has the following properties: （i） ρ（0） =0, （ii） ρ（it）=-ρ（t）,t∈C （iii） for any t∈R, ρ（t）∈R and ρ（t）≤0, and for t∈[-α, 0], ρ（t） increases from

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane,the braid monodromy factorization（and thus,the braid monodromy type）of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C2 or in CP2.In this article,we show that these groups,for the Hirzebruch surface F1,（a,b）,are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.

Calculating the Fundamental Group of Galois Cover of the(2, 3)-embedding of CP^1× T

As known to all,it is quite difficult to compute the fundamental group of a surface of general type.In this paper,applying Moishezon-Teicher’s algorithm,we investigate the fundamental group of a special surface of general type with zero topological index,namely,the Galois cover of the(2,3)-embedding of CP^1×T.Because the full presentation of the group is very complicated,we compute its special quotient and get an interesting result about its structure.

Local Phase Portraits through the Newton Diagram of a Vector Field

The Newton diagram and, in particular, the lowest-degree quasi-homogeneous terms of an analytic planar vector field allow us to determine the existence of characteristic orbits and separatrices of an isolated singular point. We give an easy algorithm for obtaining the local phase portrait near the origin of a bi-dimensional differential system and we provide several examples.

Irreducibility of The Igusa Tower

We shall give a simple （basically） the Igusa tower over Shimura varieties of PEL purely in characteristic p proof of the irreducibility of type. Our result covers Shimura variety of type A and type C classical groups, in particular, the Siegel modular varieties, the Hilbert-Siegel modular varieties, Picard surfaces and Shimura varieties of inner forms of unitary and symplectic groups over totally real fields.

ASYMPTOTIC HODGE THEORY IN SEVERAL VARIABLES:THE FLAT CASE

In the flat case, the answer to the problem posed by Steenbrink and Zucker is given.