The present paper deals with a claa of singular boundary value arising in the theory of viscous fluids ,sufficient conditions for the existence and uniqueness of positive solutions to the problems are obtained ,by employing the perturbation technique.
The classical stability of de sitter spacetime in investigated .It is found that the de sitter universe is stable, againt pertarbations of gravitational wavers ,dust matter,and the varation of the cosmological constant.
In this paper a least-squares mixed finite element methord for elasticity problems is presented and teh global estimates of mixed finite element solution are derived. The method is not object to teh classical B.-B.condition so that teh finite element space may be selected freely and optimal error estimates can be obtained.
In this paper ,teh fundementai question of teh funtion theory of several complex variables if two bounded domains are biholomorphically equivalent are studied.The symmetric classical domain can not biholomorphically equivalent to a smooth bounded strictly pseudoconvex domain espect taht itself is a smoth bounded strictly pseudoconvex domain .
This paper ,using Banach fixed point theorem,deals with an inverse problem of teh semilinear wave equation.The existence,uinquencess and stability of teh solution are given in this paper.
Some explicit and exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluidu\-t+a\-0u\-x+a\-1uu\-x+a\-2u\-\{xxx\}+b\-0u\-\{xx\}+b\-1(uu\-x)\-x+b\-2u\-\{xxx x\}=0are obtained by a kind of combiantion of the direct algebraic method and the ansatze method. These solutions include three travelling solitary wave solutions, singular travelling wave solutions and periodic wave solutions of triangular function type. Some results of other papers can be regard as a special case of the result in this paper. Futhermore, some new special solutions of the equation are obtained and some mistakes and careless of other paper are corrected.
In this paper a weighted local hardy spaces on locally compact Vilenkin groups is defined, its atomic decomposition and duality properties are given. And the behavior of certain types of convolution operators on the weighted local Hardy space is studied.
Ｉｎ ｔｈｉｓ ｐａｐｅｒ，ｗｅ ｈａｖｅ ｅｘａｃｔｌｙ ｄｅｆｉｎｅｄ ｔｈｅ ｇｅｎｅｒａｌ ｆｏｒｍｕｌａｅ ｆｏｒ ｔｈｅ ｍｕｌｔｉｖａｒｉａｔｅ ｒｅｇｒｅｓｓｉｏｎ ｃｈａｉｎ ｍｏｄｅｌ ａｎｄ ｔｈｅ ｂｌｏｃｋ ｒｅｇｒｅｓｓｉｏｎ ｃｈａｉｎ ｍｏｄｅｌ ｐｒｏｐｏｓｅｄｉｎｐａｐｅｒ ［１］ａｎｄ ［２］．Ｗｈｅｎ ｔｈｅ ｒａｎｄｏｍ ｖｅｃｔｏｒ ｉｎ ｔｈｅ ｍｏｄｅｌｓ ｄｏｅｓ ｎｏｔ ｆｏｌｌｏｗ ａ ｍｕｌｔｉｖａｒｉａｔｅ ｎｏｒｍａｌ ｄｉｓｔｒｉｂｕｔｉｏｎ，ｒｅｇｒｅｓｓｉｏｎ ｃｏｅｆｆｉｃｉｅｎｔｓ ｏｆ ｂｏｔｈ ｍｏｄｅｌｓ ａｒｅ ｒｅｐｒｅｓｅｎｔｅｄ ｉｎ ｔｅｒｍｓ ｏｆ ｔｈｅ ｅｌｅｍｅｎｔｓ ｏｆｔｈｅ ｃｏｖａｒｉａｎｃｅ ｍａｔｒｉｘ．Ｔｈｏｓｅ ｅｘｔｅｎｄ ｔｈｅ ｒｅｓｕｌｔｓ ｏｆ［１］ａｎｄ［２］ｔｏｔｈｅ ｃａｓｅｓ ｏｆ ｇｅｎｅｒａｌ ｄｉｓｔｒｉｂｕｔｉｏｎｓ．Ｆｕｒｔｈｅｒ ｗｅ ｓｈｏｗ ｔｈｅ ｒｅｌａｔｉｏｎｓｈｉｐ ｂｅｔｗｅｅｎ ｔｈｅ ｃｏｅｆｆｉｃｉｅｎｔｓ ｏｆ ａ ｓｐｅｃｉａｌ ｂｌｏｃｋ ｒｅｇｒｅｓｓｉｏｎ ｃｈａｉｎ ｍｏｄｅｌ ａｎｄ ｔｈｅ ｅｌｅｍｅｎｔｓ ｏｆ ｔｈｅ ｃｏｖａｒｉａｎｃｅ ｍａｔｒｉｘ．
Ｉｎｔｈｉｓｐａｐｅｒ，ｐｓｅｕｄｏＧｒｅｅｎ＇ｓｒｅｌａｔｉｏｎｓａｒｅｉｎｔｒｏｄｕｃｅｄｔｏｔｈｅｅｎｄｏｍｏｒｐｈｉｓｍ ｍｏｎｏｉｄｏｆａｇｒａｐｈｉｎａｎａｔｕｒａｌｗａｙ．Ｔｈｅｉｒｐｒｏｐｅｒｔｉｅｓａｒｅｉｎｖｅｓｔｉｇａｔｅｄ，ｂｙｗｈｉｃｈｗｅｒｅｖｅａｌ ｅｘｐｌｉｃｉｔｌｙｔｈａｔｔｈｅｅｎｄｏｍｏｒｐｈｉｓｍ ｍｏｎｏｉｄｏｆａｇｒａｐｈｐｏｓｓｅｓｓｅｓａｐｅｃｕｌｉａｒｓｔｒｕｃｔｕｒｅ，ｎａｍｅｌｙ ａｄｕａｌｓｔｒｕｃｔｕｒｅｉｎａｓｅｎｓｅｏｆＧｒｅｅｎ＇ｓｒｅｌａｔｉｏｎｓ．
Ｉｎｔｈｉｓｐａｐｅｒ，ａｎｏｔｉｏｎｏｎ“ｕｎｉｔｓｐｈｅｒｅ”ａｎｄ“ｕｎｉｆｏｒｍｌｙｃｏｎｖｅｘ”ｉｓｉｎｔｒｏｄｕｃｅｄｉｎ ｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｅｄｓｐａｃｅｓ，ｉｔｉｓｐｒｏｖｅｄｔｈａｔｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏｎｔｈｅｏｐｔｉｍａｌ ａｐ-ｐｒｏｘｉｍａｔｅｅｌｅｍｅｎｔ．
Ｔｈｅｅｎｔｉｒｅｃｈｒｏｍａｔｉｃｎｕｍｂｅｒχe(G) ｏｆａｐｌａｎｅｇｒａｐｈ G ｉｓ ｔｈｅｓｍａｌｌｅｓｔｎｕｍｂｅｒｏｆ ｃｏｌｏｒｓａｓｓｉｇｎｅｄｔｏｔｈｅｖｅｒｔｉｃｅｓ，ｅｄｇｅｓａｎｄｆａｃｅｓｏｆG ｓｕｃｈｔｈａｔａｎｙｔｗｏａｄｊａｃｅｎｔｏｒｉｎｃｉｄｅｎｔ ｅｌｅｍｅｎｔｓｈａｖｅｄｉｆｆｅｒｅｎｔｃｏｌｏｒｓ．Ｉｎｔｈｉｓｐａｐｅｒｗｅｇｉｖｅａｃｏｍｐｌｅｔｅｃｈａｒａｃｔｅｒｉｚａｔｉｏｎｆｏｒｔｈｅｅｎｔｉｒｅｃｈｒｏｍａｔｉｃｎｕｍｂｅｒｏｆｐｌａｎｅ ｇｒａｐｈｓ G ｗｉｔｈΔ ≥|G|－２，ｗｈｅｒｅΔｉｓｔｈｅｍａｘｉｍｕｍｄｅｇｒｅｅ ｏｆ G.
Ｔｈｅｉｎｖｅｒｓｅｐｒｏｂｌｅｍｏｎｍｉｘｅｄｐｈａｓｅｗａｖｅｌｅｔｏｆｃｏｎｖｏｌｕｔｉｏｎｍｏｄｅｌｗｈｉｃｈｔａｋｅｓ ｉｍｐｏｔｅｎｔｐｏｌｅｉｎｉｎｖｅｒｓｅｐｒｏｃｅｓｓｏｆｓｅｉｓｍｉｃｒｅｃｏｒｄｓｈａｓｗｉｄｅｌｙｂｅｅｎｐａｙｅｄａｔｔｅｎｔｉｏｎｉｎ．Ｉｎ ｔｈｅｐａｐｅｒ，ｔｈｅａｌｇｏｒｉｔｈｍｄｅｓｉｇｎｅｄｉｎｔｈｅｂａｓｅｏｎｃｏｎｃｐｔｏｆｍｉｄｄｌｅｗａｖｅｌｅｔｃｏｍｂｉｎｅｓｓｔａｔｉｓｔｉｃａｌｒｅｃｕｒｓｉｖｅａｌｇｏｒｉｔｈｍ ｗｉｔｈｉｔｅｒａｔｉｏｎｆｏｒｍａｔｉｎｗｈｉｃｈｃｒｉｔｅｒｉａｆｕｎｃｔｉｏｎｉｓａｈｉｇｈｅｒｏｒｄｅｒ ｓｔａｎｄａｒｄｉｚｉｎｇｃｕｍｕｌａｎｔ．Ｕｓｉｎｇｔｈｅａｌｇｏｒｉｔｈｍｉｎｔｈｅｐａｐｅｒ，ｅｓｔｉｍａｔｉｏｎｏｆｍｉｘｅｄｐｈａｓｅｗａｖｅｌｅｔｉｓｏｂｔａｉｎｅｄ，ａｎｄｎｏｔｏｎｅｏｆａｎｔｉｗａｖｅｌｅｔ．Ｓｉｍｕｌａｔｉｏｎｓｈａｗｓｔｈａｔｔｈｅａｌｇｏｒｉｔｈｍｉｓａｖａｉｌａｂｌｅ ｔｏｔｈｅｅｓｔｉｍａｔｉｏｎｏｆｍｉｘｅｄｐｈａｓｅｗａｖｅｌｅｔｗｈｏｓｅａｎｔｉｗａｖｅｌｅｔｈａｓａｆｉｎｉｔｅｅｎｇｈｔ．
ＯｎL2[a,b], ｔｈｅｓｔｅｅｐｅｓｔｄｅｓｃｅｎｔｓｏｌｕｔｉｏｎ ，ｆｏｒＦｒｅｄｈｏｌｍｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｓKx＝ f ａｎｄ（I－λK）x＝f，ａｒｅｓｔｕｄｉｅｄ．ＵｎｄｅｒｔｈｅｃｏｎｄｉｃｔｉｏｎｏｆK ａｎｄ（I－λK）ｂｅｉｎｇｃｏｎ ｖｅｒｓｅ，ｗｅｇｅｔｔｈｅｓｅｑｕｅｎｃｅｏｆｔｈｅｓｔｅｅｐｅｓｔｄｅｓｃｅｎｔｓｏｌｕｔｉｏｎａｎｄｐｒｏｖｅｔｈｉｓｓｅｑｕｅｎｃｅｃｏｎｖｅｒｇｉｎｇ ｔｏ ｔｈｅ ｅｘａｃｔｓｏｌｕｔｉｏｎａｎｄｇｉｖｅｏｕｔｔｈｅｅｓｔｉｍａｔｉｏｎｏｆｅｒｒｏｒ．
Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｄｅｆｉｎｅｄａｋｉｎｄｏｆＤｅｎｊｏｙｔｙｐｅｎｏｎａｂｓｏｌｕｔｅｉｎｔｅｇｒａｌｉｎ Rn，ａｎｄ ｐｒｏｖｅｄｔｈｉｓＤｅｎｊｏｙｔｙｐｅｉｎｔｅｇｒａｌｉｓｅｑｕｉｖａｌｅｎｔｔｏｔｈｅB-ｉｎｔｅｇｒａｌ．ＳｏｗｅｈａｖｅｇｉｖｅｎｔｈｅＤｅｎ ｊｏｙｆｏｒｍｏｆｔｈｅ B-ｉｎｔｅｇｒａｌ．
Ｂｙｕｓｉｎｇｗｅｉｇｈｔｆａｃｔｏｒｓ，ｔｈｅｈｏｍｏｔｏｐｙｆｏｒｍｕｌａｏｆ（ｒ，ｓ）ｄｉｆｆｅｒｅｎｔｉａｌｆｏｒｍｓｗｉｔｈ ｗｅｉｇｈｔｆａｃｔｏｒｓａｎｄｔｈｅｓｏｌｕｔｉｏｎｗｉｔｈｗｅｉｇｈｔｆａｃｔｏｒｓｏｆｅｑｕａｔｉｏｎｏｎａｌｏｃａｌｃｏｎｖｅｘｄｏ ｍａｉｎｏｎａＳｔｅｉｎｍａｎｉｆｏｌｄｗｈｉｃｈｄｏｎｏｔｉｎｖｏｌｖｅｉｎｔｅｇｒａｌｏｎｂｏｕｎｄａｒｙａｒｅｏｂｔａｉｎｅｄ，ｓｏｏｎｅ ｃａｎａｖｏｉｄｃｏｍｐｌｅｘｉｔｙｅｓｔｉｍａｔｉｏｎｓｏｆｂｏｕｎｄａｒｙｉｎｔｅｇｒａｌ．Ｆｕｒｔｈｅｒｍｏｒｅ，ｂｅｃａｕｓｅｗｅｉｇｈｔｆａｃ ｔｏｒｓａｒｅｉｎｔｒｏｄｕｃｅｄ，ｔｈｅｉｎｔｅｇｒａｌｆｏｒｍｕｌａｓｗｉｔｈｗｅｉｇｈｔｆａｃｔｏｒｓｈａｖｅｍｕｃｈｆｒｅｅｄｏｍｓｉｎａｐ ｐｌｉｃａｔｉｏｎｓ．
ＩｎｔｈｉｓｐａｐｅｒｗｅｓｈａｌｌｅｓｔａｂｌｉｓｈｔｈｅｅｘｉｓｔｅｎｃｅｔｈｅｏｒｅｍｏｆｐｏｓｉｔｉｖｅｒａｄｉａｌｌｙｓｙｍｍｅｔｒｉｃｓｏｌｕｔｉｏｎｓｏｆＤｉｒｉｃｈｌｅｔ（Ｄｉｒｉｃｈｌｅｔ－Ｎｅｕｍａｎｎ）ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｅｑｕａｔｉｏｎΔu＋f（x，u）＝０ｉｎａｎｎｕｌａｒｄｏｍａｉｎ．
ＴｈｉｓｐａｐｅｒｇｉｖｅｓａｎｅｗａｎｄｓｉｇｎｉｆｉｃａｎｔｅｘｔｅｎｓｉｏｎｏｆＨａｒｄｙＬｉｔｔｌｅｗｏｏｄＰｏｌｙａｉｎｅｑｕａｌｉｔｙｗｈｉｃｈｉｓｄｉｆｆｅｒｅｎｔｔｈａｎＨｕ ［１］．
Ｃｏｌｌｏｃａｔｉｏｎｍｅｔｈｏｄｓａｒｅｇｉｖｅｎｔｏｔｒｅａｔｔｈｅｓｅｍｉｃｏｎｄｕｃｔｏｒｐｒｏｂｌｅｍ ｍａｄｅｕｐｏｆｅｌｅｃｔｒｏｎａｎｄｈｏｌｅｃｏｎｃｅｎｔｒａｔｉｏｎｅｑｕａｔｉｏｎｓ，ａｎｄｏｐｔｉｍａｌｏｒｄｅｒｅｒｒｏｒｅｓｔｉｍａｔｅｓｉｎL ｒｅｄｅｍｏｎｓｔｒａｔｅｄ．Ｓｉｎｃｅｔｈｅｒｅａｒｅｎｏｑｕａｄｒａｔｕｒｅｓｔｏｅｖａｌｕａｔｅｔｈｅｃｏｅｆｆｉｃｉｅｎｔｓ，ｔｈｅｃｏｌｌｏｃａｔｉｏｎｍｅｔｈｏｄｒｕｎｓｎｏｔｉｃｅａｂｌｙｆａｓｔｅｒｏｎｔｈｅｃｏｍｐｕｔｅｒｔｈａｎ Ｇａｌｅｒｋｉｎ，ｇｉｖｅｎｅｘａｔｌｙｔｈｅｓａｍｅｎｏｄｅｓ．
ＵｓｅｉｓｍａｄｅｏｆｔｈｅＲａｄｏｎｔｒａｎｓｆｏｒｍｉｎｖｅｒｓｉｏｎｆｏｒｍｕｌａａｎｄＴｃｈｅｂｙｃｈｅｆｆｐｏｌｙｎｏｍｉａｌｓ，Ｔｃｈｅｂｙｃｈｅｆｆｔｒａｎｓｆｏｒｍｐａｉｒａｒｅｏｂｔａｉｎｅｄ，ｔｈｅｎｔｈｅｎｕｍｅｒｉｃａｌｉｎｖｅｒｓｉｏｎｍｅｔｈｏｄｉｓｄｅｒｉｖｅｄ．
tWe define the fatou set quasi rational mappings , to which some results of rational mappings are generalized .We also give some problems to be studies.
Ｉｎｔｈｉｓｐａｐｅｒ，ａｐｐｌｙｉｎｇｔｈｅｍａｔｈｏｄｓｏｆｐｒｏｂａｂｉｌｉｔｙａｐｐｒｏｘｉｍａｔｉｏｎｔｏｆｕｎｃｔｉｏｎｓｂｅ ｓｔｕｄｉｅｄ．Ｄｅｎｏｔｅｂｙｆｅｌｌｅｒｏｐｅｒａｔｏｒｓａｐｐｌｉｅｄｔｏｆｕｎｃｔｉｏｎ犳ａｎｄｂｙｔｈｅｔｏｔａｌｖａｒｉａｔｉｏｎｏｆｏｒｄｅｒ≥１ｏＩｔｉｓｏｂｔａｉｎｅｄｔｗｏｑｕａｎｔｉｔａｔｉｖｅｅｓｔｉｍａｔｅｔｈｅｏ ｒｅｍｓｏｎｔｈｅａｐｐｒｏｘｉｍａｔｉｏｎｄｅｇｒｅｅｏｆｆｅｌｌｅｒｏｐｅｒａｔｏｒｓｆｏｒｆｕｎｃｔｉｏｎｓｏｆｂｏｕｎｄｅｄｖａｒｉａｔｉｏｎｏｆ ｏｒｄｅｒ．
Ａｐｐｌｙｉｎｇｔｈｅｃａｌｃｕｌａｔｉｖｅｐｒｏｐｅｒｔｙｏｆｍｏｍｅｎｔｇｅｎｅｒａｔｉｎｇｆｕｎｃｔｉｏｎａｎｄｐｒｏｂａｂｉｌｉｓｔｉｃｍｅｔｈｏｄｓ，ｉｎｔｈｅｐａｐｅｒｔｈｅｃｏｎｖｅｒｇｅｎｃｅａｎｄｅｓｔｉｍａｔｉｏｎｏｆａｐｐｒｏｘｉｍａｔｉｏｎｄｅｇｒｅｅｔｏｕｎｂｏｕｎｄｅｄｆｕｎｃｔｉｏｎｓｆｏｒｇｅｎｅｒａｌｉｚｅｄＦｅｌｌｅｒｏｐｅｒａｔｏｒｓａｒｅｏｂｔａｉｎｅｄ
ＵｓｉｎｇＮｅｖａｎｌｉｎｎａｔｈｅｏｒｙｏｆｔｈｅｖａｌｕｅｄｉｓｔｒｉｂｕｔｉｏｎｏｆｍｅｒｏｍｏｐｈｉｃｆｕｎｃｔｉｏｎｓ，ｗｅ ｓｔｕｄｙｔｈｅｍｅｒｏｍｏｐｈｉｃｓｏｌｕｔｉｏｎｓｏｆｏｎｔｙｐｅｏｆｓｙｓｔｅｍｓｏｆａｌｇｅｂｒａｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ Ω１１／Ω１２ ＝ R１（z，w１，w２）， Ω２１／Ω２２ ＝ R２（z，w１，w２ ）， ａｎｄｏｂｔａｉｎｔｗｏｔｈｅｏｒｅｍｓ．
ＡｐｐｌｙｉｎｇｔｈｅｃｏｎｓｔｒｕｃｔｉｎｇｍｅｔｈｏｄｓｏｆＫａｎｔｏｒｏｖｉｃｈｔｙｐｅｏｐｅｒａｔｏｒ，ａｋｉｎｄｏｆｎｅｗｇｅｎｅｒａｌｉｚｅｄ ＷＢｅｒｎｓｔｅｉｎＫａｎｔｏｒｏｖｉｃｈｏｐｅｒａｔｏｒｈａｓｂｅｅｎｃｏｎｓｔｒｕｃｔｅｄｂｙｃｏｍｂｉｎｉｎｇａｋｉｎｄｏｆｎｅｗｂａｓｉｃＫａｎｔｏｒｏｖｉｃｈｏｐｅｒａｔｏｒｗｉｔｈｇｅｎｅｒａｌｉｚｅｄ ＷＢｅｒｎｓｔｅｉｎｏｐｅｒａｔｏｒ．ＩｔｓｃｏｎｖｅｒｇｅｎｃｅｈａｓｂｅｅｎｐｒｏｖｅｄａｎｄｉｔｓＭａｍｅｄｏｖａｓｙｍｐｔｏｔｉｃｆｏｒｍｕｌａｈａｓｂｅｅｎｐｒｅｓｅｎｔｅｄ
Ｉｎｔｈｉｓｐａｐｅｒ，ｔｈｅｃｏｎｃｅｐｔｏｆｒｅｇｕｌａｒｉｚｅｄｒｅｓｏｌｖｅｎｔｆａｍｉｌｉｅｓｉｓｉｎｔｒｏｄｕｃｅｄ，ａｎｄｉｔｓｐｒｏｐｅｒｔｉｅｓａｎｄｔｗｏｃｈａｒａｃｔｅｒｉｚａｔｉｏｎｓａｒｅｓｈｏｗｅｄ，ｗｈｉｃｈｇｅｎｅｒａｌｉｚｅｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇｒｅ ｓｕｌｔｓｏｆｒｅｓｏｌｖｅｎｔｆａｍｉｌｉｅｓｏｆｏｐｅｒａｔｏｒｓ．
ＷｅｓｈｏｗｔｈａｔR（X）：＝ｓｕｐ｛ｌｉｍｉｎｆ→ ∞R（X）：＝ｓｕｐ｛ｌｉｍｉｎｆ狀→∞‖X１＋Xn‖：‖Xn‖≤１，Xn→ ０｝＜２２ａｎｄｉｖｅｔｈａｔｔｈｅｃｏｎｄｉｔｉｏｎｔｈａｔｈａｓＷ．Ｆ．Ｐ．Ｐｆｏｒａｎｙｒｅｎｏｒｍｉｎｇ ‖．‖ ｏｆ，ｗｈｅｒｅｉｓａ Ｂａｎａｃｈｓｐａｃｅａｎｄ ｗｉｔｈｐｒｏｐｅｒｔｙ．
Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｓｔｕｄｙｔｈｅｃｏｎｖｅｒｇｅｎｃｅｓａｎｄｇｒｏｗｔｈｆｏｒｒａｎｄｏｍ Ｄｉｒｉｃｈｌｅｔｓｅｒｉｅｓ ｉｎｔｈｅｒｉｇｈｔｈａｌｆｐｌａｃｅ
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