The Uniqueness and Approximate Solution of the Positive Solution of the Uniform u_0-convex Operator Equation

M.A.Krasnosel′skii [1] proposed that "under what conditions does the solution of convex operator equation Ax=x exist and unique?".Because problem itself is more difficult, its development is quite slow.Lately Guo Da-Jun[2]suggested explicitiy that "under what conditions does u_0-convex operator have unique positive fixed point?".     

On the Equation and the Number of Solution of Znam's Problem

§1. In 1972,St.Znam posed the problem whether for every s>1 thereexist integers x_i>1,i=1,…,s such that x_i is a proper divisor of the numberx_1…x_(i-1)x_(i 1)…x_s 1 for i=1,…, s.Without loss of generality, we may assume1相似文献   

Frequency-Scale Frames and the Solution of?the?Mexican Hat Problem

We resolve a long-standing question on L ^p completeness of the time-scale (or wavelet) system generated by the Mexican hat function, when p≥2.     

A “Discretization” Technique for the Solution of ODEs II

A functional analytic technique was recently presented for finding discrete equivalent counterparts of initial value problems of ODEs and obtaining their real analytic solutions. In the current paper, this technique is extended to boundary value problems of ODEs and to the complex solutions of ODEs. In order to demonstrate this technique, it is applied to the classic Blasius problem of fluid mechanics. Apart from its real solution, its complex solution is also studied. The obtained results indicate that the complex Blasius function exhibits an oscillatory behavior and strengthen a conjecture regarding its singularities in the complex plane.     

Existence and Uniqueness of the Solution of Nonlinear Population Evolution Equations

ＥｘｉｓｔｅｎｃｅａｎｄＵｎｉｑｕｅｎｅｓｓｏｆｔｈｅＳｏｌｕｔｉｏｎｏｆＮｏｎｌｉｎｅａｒＰｏｐｕｌａｔｉｏｎＥｖｏｌｕｔｉｏｎＥｑｕａｔｉｏｎｓＬｉＨｏｎｇｙｉ（李红裔）（Ｄｅｐｔ．ｏｆＢａｓｉｃＳｃｉｅｎｃｅ，ＮｏｒｔｈＣｈｉｎａＩｎｓｔｉｔｕ...     

On the Solution of the Feigenbaum's Functional Equation

（陈芳跃）ＯｎｔｈｅＳｏｌｕｔｉｏｎｏｆｔｈｅＦｅｉｇｅｎｂａｕｍ＇ｓＦｕｎｃｔｉｏｎａｌＥｑｕａｔｉｏｎ￥ＣｈｅｎＦａｈｇｙｕｅ（Ｄａｐｔ．ｏｆＭａｔｈ．，ＺｈｅｊｉａｎｇＮｏｒｍａｌＵｎｉｖ．，Ｊｉｎｈｕａ，Ｚｈｅｊｉａｎｇ，３２１００４）Ｉｎｔ...     

Graph homomorphisms,the Tutte polynomial and “q-state Potts uniqueness”

We establish for which weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, answering a question of Freedman, Lovász and Schrijver.We introduce a new property of graphs called “q-state Potts uniqueness” and relate it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”, recently studied by Duan, Wu and Yu.     

On Existence of the Maximum Solution and Estmates on Moments of Absolutes Maximum of the Solution for Poisson S．D．E in the Plane

ＯｎＥｘｉｓｔｅｎｃｅｏｆｔｈｅＭａｘｉｍｕｍＳｏｌｕｔｉｏｎａｎｄＥｓｔｍａｔｅｓｏｎＭｏｍｅｎｔｓｏｆＡｂｓｏｌｕｔｅｓＭａｘｉｍｕｍｏｆｔｈｅＳｏｌｕｔｉｏｎｆｏｒＰｏｉｓｓｏｎＳ．Ｄ．ＥｉｎｔｈｅＰｌａｎｅＺｈａｎｇＳｕｊｉｎｇ（张素静）Ａｂ...     

A brief note on analyticity and causality,and the “levels of physics”

We argue that the causality of physics has its roots in the analyticity within the two dimensions of a fractal quantum path and further show how this picture has convergence with quantum superstrings.     

Puzzle and Solution:Diagonal String

“Suppose there were one nail in each of thefour walls of this room and in addition one nail onthe ceiling and one in the floor,Suppose furtherthat we had to tie strings between these nails.Ihave tWO colors of strings which I use—red andblue.Each connection between any two nails iseither with a red string or with a blue string. “All these strings make up many triangles,that is,any three nails can be considered apexesof a triangle formed by the strings connecting these three nails.My problem is to see if I could distribute the colors so that no triangle has all three sides the same color.”Said young Nicho—las. “This is rather complicated,”mused the mathematician. “It must involve calculations of permutations and combinations,and the like.I didn't think you knew that much algebra,Nick.”“I don’t,”replied young Nick。“but I can still do the problem.”“Well,all right,tell us how。”said the eld—er. “It is really very simple.”said young Nicho—las,“You only have to know enough to start rea-soning, “The answer is that there will be at least onetriangle all sides of which are the same color.Iwill show that it is impossible to avoid this. “Consider any one nail. Out from it theremust stretch five strings,one to each of the otherfive nails。No matter how you distribute the col—ors in these five strings,at least three of themmust be the same,since you have only two colors.For the sake of argument let assume that three ofthe strings are red. “Consider now the triangle formed betweenthree nails at which the three red strings haveterminated. “If we are to try to avoid a triangle with all three sides the same color,it follows thatthese three nails cannot all be joined to each other with one color.Putting this more simply,thetriangle formed by the three terminal nailsshould not be all blue.At least one of the stringsbetween the three nails must be red.But if so,we have completed a red triangle from the origi—nal nail.”     

Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x ²,…,x ^p ,exp?(±wx),xexp?(±wx),…,x ^m exp?(±w x)} and the second is based on the exact integration of the functions of the form: {1,x,x ²,…,x ^p ,exp?(±wx),exp?(±2wx),…,exp?(±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrödinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrödinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrödinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrödinger equation. Finally, numerical results for the resonance problem of the radial Schrödinger equation are presented.     

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Solution of Some Weight Problems for the Riemann–Liouville and Weyl Operators

The necessary and sufficient conditions are found for the weight function v, which provide the boundedness and compactness of the Riemann–Liouville operator R from L ^p to . The criteria are also established for the weight function w, which guarantee the boundedness and compactness of the Weyl operator W from to L ^q.     

Commons dilemma and the “Tragedy of the Commons”

There are a lot of studies on commons dilemma. But few studies succeed to formulate it adequately. Moreover, although there are some situations where the dilemma does not occur, these studies focuses only the case that the dilemma does occur. In the present paper, we formulate the commons dilemma as an N‐person game and examine what situations our formulation represents. We show that there are six regions with respect to the value decrement induced by one additional animal and that commons dilemma does not occur in some of these regions.     

Solution of the linear and nonlinear advection–diffusion problems on a sphere

Various linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite-volume method of second-order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator splitting allows us to develop the direct (noniterative) and fast numerical algorithm. The split problems in the longitudinal direction are solved using the Sherman-Morrison formula and Thomas algorithm. The direct solution of the split problems in the latitudinal direction requires the use of the bordering method for a block matrix, and the preliminary determination of the solution at the poles. The resulting systems with tridiagonal matrices are solved by the Thomas algorithm. The numerical experiments demonstrate that the method correctly describes the local advection–diffusion processes on the sphere, in particular, through the poles, and accurately simulate blow-up regimes (unlimited growing solutions) of nonlinear combustion, the propagation of nonlinear temperature and spiral waves, and solutions to Gray-Scott reaction–diffusion model.     

Adaptive Wavelet Solution to the Stokes Problem

This paper deals with the design and analysis of adaptive wavelet method for the Stokes problem. First, the limitation of Richardson iteration is explained and the multiplied matrix M0 in the paper of Bramble and Pasciak is proved to be the simplest possible in an appropiate sense. Similar to the divergence operator, an exact application of its dual is shown; Second, based on these above observations, an adaptive wavelet algorithm for the Stokes problem is designed. Error analysis and computational complexity are given; Finally, since our algorithm is mainly to deal with an elliptic and positive definite operator equation, the last section is devoted to the Galerkin solution of an elliptic and positive definite equation. It turns out that the upper bound for error estimation may be improved.     

A note on “On Pareto optima, the Fermat-Weber problem, and polyhedral gauges”

Received September 3, 1997 / Revised version received March 20, 1998 Published online October 9, 1998