A Class of Quasi-Linear Riemann-Hilbert Problems for General Holomorphic Functions in the Unit Disk

In this paper, a class of quasi linear Riemann Hilbert problems for general holomorphic functions in the unit disk was studied. Under suitable hypotheses, the existence of solutions of the Hardy class H 2 to this problem was proved by means of Tikhonov's fixed point theorem and corresponding theories for general holomorphic functions.

ON HILBERT BOUNDARY VALUE PROBLEMS WITH RADICAL

The Hilbert boundary value problem Re{λ（t） p√ψ＋（t）} = c（t）, t∈L of normal type with Holder continuous coefficients is discussed, where L is the unit circle ｜t｜ = 1,p ≥2 is any definite integer,ψ^＋（t） is the boundary value of the unknown function ψ（z） holomorphic in ｜z｜ 〈 1 with single-valued continuous p√ψ＋（t） on L.

THE HILBERT BOUNDARY VALUE PROBLEM FOR GENERALIZED ANALYTIC FUNCTIONS IN CLIFFORD ANALYSIS

Let R0,n be the real Clifford algebra generated by e1, e2,... , en satisfying eiej＋ejei=-2δij,i,j=1,2…,ne0 is the unit element.Let Ω be an open set. A function f is called left generalized analytic in ft if f satisfies the equation Lf=0,where ……qi 〉0, i =-, 1, - ……, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in Rn＋1 will be investigated.

Generalized Strongly Nonlinear Quasi-Complementarity Problems

Using the algorithm in this paper, we prove the existence of solutions to the gene-ralized strongly nonlinear quasi-complementarity problems and the convergence of theiterative sequences generated by the algorithm. Our results improve and extend thecorresponding results of Noor and Chang-Huang. Moreover, a more general iterativealgorithm for finding the approximate solution of generalized strongly nonlinear quasi-complementarity problems is also given. It is shown that the approximate solution ob-tained by the iterative scheme converges to the exact solution of this quasi-com-plementarity problem.

N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.

The Riemann Hilbert Problem for General Elliptic Complex Equations of Fourth Order

This paper deals with the existence theorem and Riemann Hilbert boundary value problem for general nonlinear elliptic complex equations of fourth order. Firstly we give the representation and existence theorem of solutions for the complex equations. Moreover,we propose the Riemann Hilbert problem and its well posedness,and then we give the representation of solutions for the modified boundary value problem and prove its solvsbility,and finally derive solvability conditions of the original Riemann Hilbert problem.

THE HILBERT BOUNDARY VALUE PROBLEM FOR ANALYTIC FUNCTION OF SEVERAL COMPLEXVARIABLES WITH DISCONTINUOUS COEFFICIENTS

This paper deals with the Hilbert boundary value problem for analytic function of several complex variables with discoutiuuous codsdents on polycylinder ring. The author gives the corresponding metamorphous problem and gets the condition of solvability and an intergral representation of the solution.

Hilbert’s First Problem and the New Progress of Infinity Theory

In the 19th century, Cantor created the infinite cardinal number theory based on the “1-1 correspondence” principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of “whole is greater than part”, and created another ruler for measuring infinite sets. The discussion in this paper shows that, compared with the cardinal number method, the Grossone method enables infinity calculation to achieve a leap from qualitative calculation to quantitative calculation. According to Grossone theory, there is neither the largest infinity and infinitesimal, nor the smallest infinity and infinitesimal. Hilbert’s first problem was caused by the immaturity of the infinity theory.

New Formulation for Arbitrary Cracks Problem and Its Stress Intensity Factor of Plane Elasticity

Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classical plane elastic crack model, only the known conditions were revised in the new formulation, which are greatly convenient to solve the problem, and no other new condition was given. Results and Conclusion The general exact analytic solution is given here based on the formulation though the problem is very complicated. Furthermore, the stress intensity factors K Ⅰ, K Ⅱ of the problem are also given.

UNDERSTANDING SCHUBERT’S BOOK(Ⅱ)

In this paper,we give rigorous justification of the ideas put forward in§20,Chapter 4 of Schubert’s book;a section that deals with the enumeration of conics in space.In that section,Schubert introduced two degenerate conditions about conics,i.e.,the double line and the two intersection lines.Using these two degenerate conditions,he obtained all relations regarding the following three conditions:conics whose planes pass through a given point,conics intersecting with a given line,and conics which are tangent to a given plane.We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert’s idea.

Matrix integrable fifth-order mKdV equations and their soliton solutions

We consider matrix integrable fifth-order mKdV equations via a kind of group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on properties of eigenvalue and adjoint eigenvalue problems, we solve the corresponding Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues, and construct their soliton solutions, when there are zero reflection coefficients. Illustrative examples of scalar and two-component integrable fifthorder mKdV equations are given.

UNDERSTANDING SCHUBERT’S BOOK(Ⅰ)

Hilbert Problem 15 required an understanding of Schubert’s book[1],both its methods and its results.In this paper,following his idea,we prove that the formulas in§6,§7,§10,about the incidence of points,lines and planes,are all correct.As an application,we prove formulas 8 and 9 in§12,which are frequently used in his book.

Stresses due to an adhesion crack in T-shaped junction of two orthotropic plates

The general solution of stresses is derived for a T-shaped junction of two thin plates with an adhesion crack. The plates are orthotropic. A shear force is applied on the crack surface. The analysis is based on the supposition that the stresses in each plate can be approximated by a plane stress condition. The results obtained are verified by numerical calculation of FEM.

UNDERSTANDING SCHUBERT'S BOOK(Ⅲ)

In§13 of Schubert’s famous book on enumerative geometry,he provided a few formulas called coincidence formulas,which deal with coincidence points where a pair of points coincide.These formulas play an important role in his method.As an application,Schubert utilized these formulas to give a second method for calculating the number of planar curves in a one dimensional system that are tangent to a given planar curve.In this paper,we give proofs for these formulas and justify his application to planar curves in the language of modern algebraic geometry.We also prove that curves that are tangent to a given planar curve is actually a condition in the space of planar curves and other relevant issues.

The Solution of A Compound Periodic Boundary Problem

We have studied the compound periodic boundary problem in the upper halfplane above the real axis. Under proper conditions, we obtain a periodic and sectionallyholo-morphic function in the upper half plane. In addition, we have also solved the compoundboundary problem with discontinuities of the first kind of the coefficients in the Hilbertcondition.

The Historical Logic on the Basic Theory of Physics—A Summary on the Cosmic Continuum Theory

Any scientific system has a unified basic theory. But physics has no unified basic theory in the modern sense. Classical mechanics, relativity and quantum mechanics have their own basic concepts, categories and principles, so none of them can be regarded as true basic theories of physics. Cosmic Continuum Theory holds that the continuity and discreteness of the universe are fundamental issues related to the unification of physics. Because the contradiction between quantum non-locality and local reality is the fundamental obstacle to the unification of physics, while locality and non-locality correspond to the continuity and discreteness of physical reality respectively. The cosmic continuum theory introduces mathematical continuum and axiomatic ideas to reconstruct the basic theory of physics, and by the correspondence of existence and its dimensions to achieve the unification of the essence of physical reality, by introducing the cosmic continuum hypothesis to achieve the unification of the continuity and discreteness of physical reality, by introducing axiomatic methods to achieve formal unification of the foundations on physics. From the perspective of Cosmic Continuum, classical mechanics, relativity and quantum mechanics are no longer the basic theories of physics, but three branch theories of physics that are respectively applicable to macroscopic, cosmoscopic and microcosmic systems.

Reconstruction method of differentiated backprojection-projection onto convex sets in the interior problem and design of bone-nail model

This work focuses on the application of the reconstruction method of differentiated backprojection （DBP）-projection onto convex sets （POCS） in the interior problem.First,we present the definition of the interior problem and real truncated Hilbert transform,and then outline the implementation steps of DBP-POCS.After that,we introduce the middle-part known condition for region of interest （ROI） accurate reconstruction and the unique condition of the interior problem,and verify the uniqueness and stability of the interior problem accurate reconstruction through numerical experiments,and then compare the results for the interior problem in reconstruction images using filtered backprojection （FBP）.In addition,the authors also design the application models of ROI reconstruction and make an initial attempt to the application of DBP-POCS method in the interior problem.

Canards Flying on Bifurcation

There exists a property “structural stability” for “4-dimensional canards” which is a singular-limit solution in a slow-fast system with a bifurcation parameter. It means that the system includes the possibility to have some critical values on the bifurcation parameter. Corresponding to these values, the pseudo-singular point, which is a singular point in the time-scaled-reduced system should be changed to another one. Then, the canards may fly to another pseudo-singular point, if possible. Can the canards fly? The structural stability gives the possibility for the canards flying. The precise reasons why happen are described in this paper.

Structural Stability in 4-Dimensional Canards

Let us consider higher dimensional canards in a sow-fast system R2+2 with a bifurcation parameter. Then, the slow manifold sometimes shows various aspects due to the bifurcation. Introducing a key notion “symmetry” to the slow-fast system, it becomes clear when the pseudo singular point obtains the structural stability or not. It should be treated with a general case. Then, it will also be given about the sufficient conditions for the existence of the center manifold under being “symmetry”. The higher dimensional canards in the sow-fast system are deeply related to Hilbert’s 16th problem. Furthermore, computer simulations are done for the systems having Brownian motions. As a result, the rigidity for the system is confirmed.

ON THE CONNECTION BETWEEN TWO PARTS OF HILBERT′S 16TH PROBLEM AND EQUIVARIANT BIFURCATIONP ROBLEM

This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the periodic solution family of ( m-1) degree planar Hamiltonian systems with Z q equivariant (or D q equivariant) symmetry to realize some schemes of ovals for planar algebraic curves? On the contrary, if an algebraic curve of degree m has maximal number of branches of ovals (it is called M -curve), can we make his all ovals become limit cycles of a planar polynomial system? What schemes of distribution of limit cycles can be realized by polynomial system.