On Spirallike Mappings in Several Complex Variables

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ１９３３,Ｈ．ｃａｒｔａｎｐｏｉｎｔｅｄｏｕｔｔｈａｔｔｈｅｒｅｄｏｅｓｎｏｔｅｘｉｓｔｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇＢｉｅｂｅｒｂａｃｈｃｏｎ－ｊｅｃｔｕｒｅｆｏｒｂｉｈｏｌｏｍｏｒｐｈｉｃｍａｐｐｉｎｇｓｉｎｓｅｖｅｒａｌｃｏｍｐｌｅｘｖａｒｉａｂｌｅｓ．Ｈｅａｌｓｏｐｏｉｎｔｅｄｏｕｔｔｈｅｃｏｒｒｅ－ｓｐｏｎｄｉｎｇｇｒｏｗｔｈａｎｄｃｏｖｅｒｉｎｇｔｈｅｏｒｅｍｓｆａｉｌｉｎｔｈｅｃａｓｅｏｆｓｅｖｅｒａｌｃｏｍｐｌｅｘｖａｒｉａｂｌｅｓ．Ｔｈｅｎ…     

离散的Painlevé方程Ⅱ的有理解

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＮｏｎｌｉｎｅａｒｉｎｔｅｇｒａｂｌｅｄｉｓｃｒｅｔｅｓｙｓｔｅｍｓａｒｅｎｏｗａｔｔｒａｃｔｉｎｇｍｕｃｈａｔｔｅｎｔｉｏｎ．Ａｍｏｎｇｔｈｅｍ,ｔｈｅｄｉｓｃｒｅｔｅＰａｉｎｌｅｖéｅｑｕａｔｉｏｎｓａｒｅｅｘｐｅｃｔｅｄｔｏｂｅｔｈｅｍｏｓｔｆｕｎｄａｍｅｎｔａｌｏｎｅｓｉｎａｎａｌｏｇｙｔｏｔｈｅｃｏｎｔｉｎｕｏｕｓｃａｓｅ．Ｃｌｏｓｅｒｓｔｕｄｉｅｓａｒｅｈｏｗｒｅｖｅａｌｉｎｇｔｈｅｉｒｒｉｃｈｍａｔｈｅｍａｔｉｃａｌｓｔｒｕｃｔｕｒｅｓ,ｓｕｃｈａｓ…     

Two Dimensional Submanifolds in Four Manifolds with Boundary

§０．　ＩｎｔｒｏｄｕｃｔｉｏｎＦｏｒａｃｌｏｓｅｄ４－ｍａｎｉｆｏｌｄ,ｉｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔａｎｙｔｗｏｄｉｍｅｎｓｉｏｎａｌｈｏｍｏｌｏｇｙｃｌａｓｓｃａｎｂｅｒｅｐｒｅｓｅｎｔｅｄｂｙａｎｅｍｂｅｄｄｅｄｓｕｒｆａｃｅ．Ａｆｕｎｄａｍｅｎｔａｌｐｒｏｂｌｅｍｉｎ４－ｄｉｍｅｎｓｉｏｎａｌｔｏｐｏｌｏｇｙｉｓｔｏｆｉｎｄａｓｕｒｆａｃｅｗｉｔｈｍｉｎｉｍａｌｇｅｎｕｓｗｈｉｃｈｒｅｐｒｅｓｅｎｔｓｔｈｅｇｉｖｅｎｈｏｍｏｌｏｇｙｃｌａｓｓ．Ａｓｐｅｃｉａｌｃａｓｅｏｆｔｈｉｓｑ…     

Stability Analysis of Optimal Control Systems via Liapunov Method

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｅｃｏｎｄｍｅｔｈｏｄｏｆＬｉａｐｕｎｏｖｎｅｅｄｎｏｔｓｏｌｖｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ,ｂｕｔｉｔｃａｎｄｉｒｅｃｔｌｙｇｉｖｅｓｔａｂｉｌｉｔｙｉｍｆｏｒｍａｔｉｏｎｏｆｓｙｓｔｅｍｅｑｕｉｌｉｂｒｉｕｍｓｔａｔｅｂｙｃｏｎｓｔｉｔｕｔｉｎｇＬｉａｐｕｎｏｖｆｕｎｃｔｉｏｎ．Ｉｔｉｓｖｅｒｙｕｓｅｆｕｌｆｏｒｔｈｏｓｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｈｉｃｈｃａｎｎｏｔｅａｓｉｌｙｂｅｓｏｌｖｅｄ．Ｆｏｒｇｅｎｅｒａｌｓ…     

格蕴涵代数与MV-代数

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｏｒｄｅｒｔｏｒｅｓｅａｒｃｈｔｈｅｌｏｇｉｃａｌｓｙｓｔｅｍｗｈｏｓｅｐｒｏｐｏｓｉｔｉｏｎａｌｖａｌｕｅｉｓｇｉｖｅｎｉｎａｌａｔｔｉｃｅｆｒｏｍｔｈｅｓｅｍａｎｔｉｃｖｉｅｗｐｏｉｎｔ,ｗｅｈａｖｅｐｒｏｐｏｓｅｄｔｈｅｃｏｎｃｅｐｔｏｆｌａｔｔｉｃｅｉｍｐｌｉｃａｔｉｏｎａｌｇｅｂｒａｓｉｎ［１］ａｎｄｈａｖｅｄｉｓｃｕｓｓｅｄｔｈｅｉｒｓｏｍｅｐｒｏｐｅｒｔｉｅｓ．ＭＶ－ａｌｇｅｂｒａｓｗｅｒｅｉｎｖｅｎｔｅｄｂｙＣ．Ｃ．Ｃｈａｎｇ［２］ｉｎｏｒｄｅ…     

Quasi—uniformities on Lattice Implication Algebras

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｏｒｄｅｒｔｏｒｅｓｅａｒｃｈｔｈｅｌｏｇｉｃａｌｓｙｓｔｅｍｗｈｏｓｅｐｒｏｐｏｓｉｔｉｏｎａｌｖａｌｕｅｉｓｇｉｖｅｎｉｎａｌａｔｔｉｃｅ，ＸｕＹ．［Ｘｕ２］ｐｒｏｐｏｓｅｄｔｈｅｃｏｎｃｅｐｔｏｆｌａｔｔｉｃｅｉｍｐｌｉｃａｔｉｏｎａｌｇｅｂｒａａｎｄｄｉｓｃｕｓｓｅｄｉｔｓｓｏｍｅｐｒｏｐｅｒ－ｔｉｅｓｉｎ［Ｘｕ１］ａｎｄ［Ｘｕ２］．Ａｌｓｏ，ｉｎ［ＸＱ１］，ＸｕＹ．ｔｏｇｅｔｈｅｒｗｉｔｈＫ．Ｙ．Ｑｉｎｄｉｓｃｕｓｓｅｄｔｈｅｐｒｏｐ－ｅｒｔｉｅｓｏ…     

Characterizations of Stable Functions on L—domains

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｐｕｒｐｏｓｅｏｆＤｏｍａｉｎｓｔｈｅｏｒｙ（ｉｎｔｒｏｄｕｃｅｄｂｙＤ．Ｓｃｏｔｔ）ｉｓｔｏｍｏｄｅｌｔｈｅｄｅｎｏｔａｔｉｏｎａｌｓｅ－ｍａｎｔｉｃｓｏｆｃｏｍｐｕｔｅｒｐｒｏｇｒａｍｍｏｎｇｌａｎｇｕａｇｅｓ［１］．Ａｎｉｍｐｏｒｔａｎｔｆｒａｍｅ－ｗｏｒｋｆｏｒｔｈｅｄｅｎｏｔａ－ｔｉｏｎａｌｓｅｍａｎｔｉｃｓｏｆｐｒｏｇｒａｍｍｉｎｇｌａｎｇｕａｇｅｓｉｓｔｈｅｃａｔｅｇｏｒｙｏｆＬ－ｄｏｍａｉｎｓｗｉｔｈｓｔａｂｌｅｆｕｎｃｔｉｏｎｓ．Ｌ－ｄｏｍ…     

A Property of MRA of Multiplicity r with an Arbitrary Dilation Matrix

ＩｔｉｓｗｅｌｌｋｎｏｗｎｔｈａｔｔｈｅｍｅｔｈｏｄｔｏｃｏｎｓｔｒｕｃｔｗａｖｅｌｅｔｉｓｍｕｌｔｉｒｅｓｏｌｕｔｉｏｎａｎａｌｙｓｉｓｗｈｉｃｈｗａｓｇｉｖｅｎｆｉｒｓｔｂｙＳ．Ｍａｌｌａｔ（［１］,［２］）ｉｎ１９８９．Ｉｎ１９９４,Ｔ．Ｎ．Ｔ．ＧｏｏｄｍａｎａｎｄＳ．Ｌ．Ｌｅｅ（［３］）ｇｅｎｅｒａｌｉｚｅｄｃｏｎｃｅｐｔｏｆｍｕｌｔｉｒｅｓｏｌｕｔｉｏｎａｎａｌｙｓｉｓｔｏｔｈｅｃａｓｅｏｆｍｕｌｔｉｐｈｃｉｔｙｒ（ｒ１）ｓｕｃｈｔｈａｔｍｕｌｔｉｗａｖｅｌｅｔｓｃａｎｂｅｃｏｎｓｔ…     

Analysis and Forecast on the Car Market of Our Country

§１．　ＦｏｒｅｃａｓｔａｂｏｕｔｔｈｅＣａｒＭａｒｋｅｔＤｅｍａｎｄａｎｄｔｈｅＱｕａｎｔｉｔｙｏｆＣａｒＰｏｓｓｅｓｓｉｏｎｉｎＯｕｒＣｏｕｎｔｒｙｉｎ２０００Ｉｎｔｈｉｓｓｅｃｔｉｏｎ,ｗｅｓｈａｌｌｆｏｒｅｃａｓｔｔｈｅｃａｒｍａｒｋｅｔｄｅｍａｎｄａｎｄｔｈｅｑｕａｎｔｉｔｙｏｆｃａｒｐｏｓｓｅｓｓｉｏｎｉｎｏｕｒｃｏｕｎｔｒｙｉｎ２０００ｂｙｍｅａｎｓｏｆｔｗｏｓｔａｔｉｓｔｉｃｓｍｅｔｈｏｄｓ,ｉ．ｅ．ｔｅｎｄｅｎｃｙｉｎｆｅｒｅｎｃｅａｎｄｒｅｇｒｅｓ－ｓｉｏｎａｎａｌｙｓｉｓ．…     

REALHYPERSURFACESWITHCONSTANTPRINCIPALCURVATUREINCOMPLEXHYPERBOLICSPACES

ＡｂｓｔｒａｃｔＬｅｔＭｂｅａｒｅａｌｈｙｐｅｒｓｕｒｆａｃｅｗｉｔｈｔｈｒｅｅ．ｄｉｓｔｉｎｃｔｃｏｎｓｔａｎｔｐｒｉｎｃｉｐａｌｃｕｒ－ｖａｔｕｒｅｓｉｎｔｗｏｄｉｍｅｎｓｉｏｎａｌｃｏｍｐｌｅｘｈｙｐｅｒｂｏｌｉｃｓｐａｃｅＣＨ，ｔｈｅｎｅｉｔｈｅｒ（ａ）Ｍｈａｓｔｗｏｐｒｉｎｃｉｐａｌｃｕｒｖａｔｕｒｅｓｗｈｏｓｅｄｉｆｆｅｒｅｎｃｅｉｓｌａｒｇｅｔｈａｎｚｅｒｏａｎｄｌｅｓｓｔｈａｎｏｎｅ．（ｂ）ＴｈｅｔｈｒｅｅｐｒｉｎｃｉｐａｌｃｕｒｖａｔｕｒｅｓｏｆＭｆｏｒｍａｎｅｑｕｉｄｉｆｆｅｒｅｎｃｅｓｅｑｕｅｎｃｅｗｈｏｓｅｃｏｍｍｏｎｄｉｆｆｅｒｅｎｃｅｉｓｏｎｅ；（ｃ）Ｍｉｓｈｏｌｏｍｏｒｐｈｉｃｃｏｎｇｒｕｅｎｔｔｏａｎｏｐｅｎｐａｒｔｏｆａｔｕｂｅａｒｏｕｎｄｒｅａｌｔｗｏｄｉｍｅｎｔｉｏｎａｌｈｙｐｅｒｂｏｌｉｃｓｐａｃｅ．．Ｋｅｙｗｏｒｄｓ．Ｒｅａｌｈｙｐｅｒｓｕｒｆａｃｅｃｏｍｐｌｅｘｈｙｐｅｒｂｏｌｉｃｓｐａｃｅｔｕｂｅｌｅｑｕｉｄｉｆｆｅｒｅｎｃｅｓｅｑｕｅｎｃｅ     

An inverse problem for supersonic flow past a curved wedge

This paper studies an inverse problem for supersonic potential flow past a curved wedge, in which we design a suitable curved wedge such that the shock produced by the curved wedge can be controlled to the given position. Under suitable conditions, by characteristic method, we prove the existence of the global classical solution to this inverse problem and develop an optimal decay rate on the given shock’s second order derivatives. We finally construct a specific wedge such that the shock generated by the wedge is a convex combined one.     

Supersonic flow past a concave double wedge

The supersonic flow past a concave double wedge is discussed. Because of the interaction between the outer shock attached at the head of the wedge and the inner shock issued from the concave corner, there is a rarefaction wave issued from the intersection of the outer and inner shock. The rarefaction wave is reflected by the outer shock and the wedge infinitely, while the outer shock is also bent due to interaction. The global existence of the solution is proved under the assumptions that the outer shock is weak and the difference of two slopes of the double wedge is small. Meanwhile, a rigorous proof of the asymptotic behavior of the global solution is given. The property is often ap plied to numerical computation. Project partially supported by the National Natural Science Foundation of China and Doctoral Programme Foundation of IHEC.     

Justification of the Courant-Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

As is well known, two solutions of the problem of a supersonic stationary inviscid nonheatconducting gas flow onto a planar infinite wedge are theoretically possible: the solution with a strong shock (the flow speed behind the shock is subsonic) and the solution with a weak shock (the flow speed behind the shock is supersonic). Unlike the well-studied case of a strong shock that is generically unstable [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Study of the stability in the problem on flowing around a wedge. The case of strong wave, J. Math. Anal. Appl. 319 (2006) 248-277; A.M. Blokhin, D.L. Tkachev, Yu.Yu. Pashinin, Stability condition for strong shock waves in the problem of flow around an infinite plane wedge, Nonlinear Anal. Hybrid Syst. 2 (2008) 1-17], R. Courant and K.O. Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948] assumed that the solution with a weak shock is asymptotically stable by Lyapunov. Presentation of classical solution to the corresponding problem which is found in the present paper is the first step on the way to justification of Courant-Friedrichs hypothesis on linear level.     

Supersonic flow onto a solid wedge

We consider the problem of two‐dimensional supersonic flow onto a solid wedge, or equivalently in a concave corner formed by two solid walls. For mild corners, there are two possible steady state solutions, one with a strong and one with a weak shock emanating from the corner. The weak shock is observed in supersonic flights. A longstanding natural conjecture is that the strong shock is unstable in some sense. We resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed. More precisely, we prove that for upstream state as initial data in the entire domain, the time‐dependent solution is self‐similar, with a weak shock at the tip of the wedge. We construct analytic solutions for self‐similar potential flow, both isothermal and isentropic with arbitrary γ ≥ 1. In the process of constructing the self‐similar solution, we develop a large number of theoretical tools for these elliptic regions. These tools allow us to establish large‐data results rather than a small perturbation. We show that the wave pattern persists as long as the weak shock is supersonic‐supersonic; when this is no longer true, numerics show a physical change of behavior. In addition, we obtain rather detailed information about the elliptic region, including analyticity as well as bounds for velocity components and shock tangents. © 2007 Wiley Periodicals, Inc.     

Steady supersonic flow past an almost straight wedge with large vertex angle

This paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary. It is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar, the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic. This paper is devoted to generalizing this result. Under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small, a sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to a global weak solution of the steady problem. A sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution. The regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Study of the stability in the problem on flowing around a wedge. The case of strong wave

We consider the flow of an inviscid nonheatconducting gas in the thermodynamical equilibrium state around a plane infinite wedge and study the stationary solution to this problem, the so-called strong shock wave; the flow behind the shock front is subsonic.We find a solution to a mixed problem for a linear analog of the initial problem, prove that the solution trace on the shock wave is the superposition of direct and reflected waves, and, the main point, justify the Lyapunov asymptotical stability of the strong shock wave provided that the angle at the wedge vertex is small, the uniform Lopatinsky condition is fulfilled, the initial data have a compact support, and the solvability conditions take place if needed (their number depends on the class in which the generalized solution is found).     

Continuous compression waves in the two-dimensional Riemann problem

The interaction between a plane shock wave in a plate and a wedge is considered within the framework of the nondissipative compressible fluid dynamic equations. The wedge is filled with a material that may differ from that of the plate. Based on the numerical solution of the original equations, self-similar solutions are obtained for several versions of the problem with an iron plate and a wedge filled with aluminum and for the interaction of a shock wave in air with a rigid wedge. The behavior of the solids at high pressures is approximately described by a two-term equation of state. In all the problems, a two-dimensional continuous compression wave develops as a wave reflected from the wedge or as a wave adjacent to the reflected shock. In contrast to a gradient catastrophe typical of one-dimensional continuous compression waves, the spatial gradient of a two-dimensional compression wave decreases over time due to the self-similarity of the solution. It is conjectured that a phenomenon opposite to the gradient catastrophe can occur in an actual flow with dissipative processes like viscosity and heat conduction. Specifically, an initial shock wave is transformed over time into a continuous compression wave of the same amplitude.     

THE PROBLEM FOR THE SUPERSONIC PLANE FLOW PAST A CURVED WEDGE

The problem for the supersonic plane flow described by TSD equation past a curved wedge is considered.For a given curved wedge,we will determine the corresponding shock and the solution behind the shock.Moreover,under suitable assumptions,we obtain the global existence and uniqueness for the above mentioned problem.     

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317-318): when a supersonic flow hits a sharp wedge, it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined.