Jensen型方程的HyersUlam稳定性

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广义函数空间上Quadratic函数方程的Hyers-Ulam-Rassias型稳定性

ｅ
≤
ｚＩｘ）ｄ＋
＞
ｄｚ≤ Ｃｅ．
引理２设（）是非负连续Ｐ阶齐次函数，中０＜Ｐ＜２ｚ其．若
（ｆ，）一（＊Ｅ）ｚ）一（
Ｊ（ｙ（ｔ则任￡ｏ级 ∑ 。ｘ）ｙ），对意＞，数２ｍＥ，ｄ
首先给出几个记号：一（，，ｚ … ｚ）∈ Ｒ，ａ一（，，）∈ Ｎ，一ｚ … ．：，一 … ａｚｚａ … ，
ａ一ａａ／ｘ，其中表示非负整数集．
定义若
∈
上的无穷可微函数，Ｖ，对卢∈ Ｎ，足不等式满
（ｏ２（，）一＜，（，）ｚ． “ Ｐ，ｚ）Ｉｚｄ）
下面我们引入热方程的核Ｅ（）一Ｅ，（）一（丌）ｘ（１１／￡，ｔ０４￡～ｅｐ一４）＞．它的半群性质
（（）一Ｅ（）在本文中具有重要作用．义函数 ∈ Ｅ＊Ｅ）ｚ一ｚ广
Ｙｌ≥
（）３
其中Ｍ与Ｊ关．级数∑ ２ ’ （，２在无故。ｔｊ任意紧一敛．式（）理１引／）集上致收再由３及引，理２
ＵｌｍＲｓｉｓｓａｉｔｆｕｄａｉｆｎｔｎｌｑａｉｎｉｉｒｕｉｎ，ｉｅｅａｚｄｔｅｒｓｌｏ８，ａ — ａｓｔｂｌｙｏａｒｔｕｃｉａｅｕｔｄｓｉｔｓｗｈｃｇｎｒｌｅｈｅｕｔｆ］ａｉｑｃｏｏｎｔｂｏｈｉＥ

向量值α次熵映射及熵方程的Hyers-Ulam稳定性

＆次熵映射序列是稳定的，并给出向量值Ｏ次熵映射序列的一般形式．ｚ
本文用表示全体实数之集，表示全体正整数之集，表示数域 Ⅲ＝或ＣＮ＇（）上的任一赋范
线性空间，ｘ∈Ｘ及 ∈Ｆ约定：Ｖ，＝．定义
Ｄ＝｛，） Ⅱ ，０１，（Ｙ ∈ ：Ｙ∈【），＋Ｙ≤１）
ｒ＝ｐ ∈”Ｐ＝）… ∑ １， ≥，｛Ｌ（：≥ １１ｉ＝Ｊ礼２ ∈ …，）ｉ２｝佗＝１
在ｒ上定义函数
ｌ（一一）（１ ≠１２。１ ∑ｐ一）一，群（，，）ｌｐ … Ｐ＝｛１
第１卷第２３期
２１ＩＪＩＹＳＳＦＵＮＣＴＩＯＮＡＬＩＳＡＰＰＬＩＣＡＴＡ
Ｖｏ１１３．Ｎｏ．．２
Ｊｎ，０１ｕｅ２１
ＤＯ１．２／ＰＪ１０２１．１８Ｉ：０７４Ｓ．１６．１０８３．００文章编号：１０—３７２１）２１８—８０９１２（０１０— １８００
（．）２４
（，）（，）１０：０１则称｛）是２半对称的．一
例２１由文献［６知：Ｘ＝且０．５］当－＜乜≠１厶＝ＨＺ满足上述（１一２）时，２）（．．．４定义２２对映射ｇ：，卜＋及正数Ｏ定义映射Ｅｇ： — 为．［１０Ｌ，Ｄ
２若ｖｐ，２Ｐ）３有）（１Ｐ，３ ∈Ｆ，（ｌＰ，３：／（ｌＰ，２Ｐ，２）３Ｐ，３Ｐ）Ｐ
则称映射列｛厶）是３半对称的；一
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拟Banach代数上拟同态的Hyers-Ulam—Rassias稳定性(Ⅰ)

究．－ｉｃｏｘｓｎ．ｄ．ｎＥｍａ：ａｈ￣ｎｕｅｕｃｌ
４２２
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织
高
校
基
础
科
学
学
报
第２卷１
（）Ｖ ∈ Ｒ，２Ｘ∈ Ｘ，ＩｌＩｌ１．ｌＩ＝ＩＩ１ｚ（）ｊ ≥ １得ＶＹ∈ ｘ，Ｉ＋ＹＪＫ（Ｉ３Ｋ使，Ｉｌ≤ ＪＩＩ）Ｉ＋ＩＩ．ＸＩＹ若ＩＩ是Ｘ上的拟范数，Ｉ・ｌ称序对（ＩＩＸ，Ｉ・ｌ）为拟赋范空间．最小的Ｋ称为ｌｌ的凹性模．１・Ｉ拟
（）ＶＸＥＸ，１０ｆ０且｝ｚｌ≥ Ｉｆ０当且仅当ｚ一０ｆ一ｚ．
收稿日期：０８０ — ４２０ — ５０
基金项目：国家自然科学基金资助项目（０７１３１５１１）
通讯作者：曹怀信（９８）男，１５一，陕西省长武县人，陕西师范大学教授，博士．主要从事算子理论与小波分析等方面的研
文章编号：０６８４（０８０ —４１０１０ —３１２０）４０２ —４
拟Ｂｎｃａａｈ代数上拟同态的Ｈｙｒ～ａＲａｓｓ定性（ｅｓＵｌｍ— ｓｉ稳ａ工）
郭艳平，怀信曹
（西师范大学数学与信息科学学院，陕陕西西安７０６）１０２
中图分类号：７．Ｏ１７１
文献标识码：Ａ
１引言和定义
文献［］于１４１９０年首先提出了环同态的稳定性问题：设Ｇ是一个群，ｚＧ是一个带有距离（，・・）的度量群，映射厂：Ｇ．＞０问是否ｊＧ一Ｖ￡，＞０当，ｄｆｘ）厂）（）（（ｙ，（厂） ≤ （，时，在群同态ｇＧ一Ｇ，ＶｚＹＥＧ）存：－２使得ｄｆ（，（） ≤ ￡（（）ｇｚ）Ｖｚ∈ Ｇ）现在，？这个问题已经被许多的学者给出了确切的回答．人们称如此的一个映射，：。Ｇ为近似同Ｇ一

非线性微分方程的Hyers-Ulam稳定性

Ｈｙｅｒｓ — Ｕｌａｍｓｔａｂｉｌｉｔｙｏｆｎｏｎｌｉｎｅａｒｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ
ＬＩＹｏｎｇｊｉｎ，ＨＵＡＬｉｕｂｉｎ
（１．ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｕｎＹａｔ－ｓｅｎＵｎｉｖｅｒｓｉｔｙ，Ｇｕａｎｇｚｈｏｕ５１０２７５，Ｃｈｉｎａ；
ｌｙｙ一１Ｉ ≤占，ａｎｄＹ（ｎ）＝１，ｔｈｅｎｔｈｅｒｅｅｘｉｓｔｓ ∈Ｃ［ Ⅱ ，ｂ］ｗｉｔｈ一１＝０ａｎｄ（）＝１，ｓｕｃｈｔｈａｔ
Ｉｙ（ｘ）一ｚ（ｘ）Ｉ＜Ｋ６．
［Ｋｅｙｗｏｒｄｓ］Ｈｙｅｒｓ－Ｕｌａｍｓｔａｂｉｌｉｔｙ；ｎｏｎｌｉｎｅａｒｄｉｆｆｅｒｅｎｔｉｌａｅｑｕａｔｉｏｎ；ｉｎｉｔｉａｌｃｏｎｄｉｔｉｏｎｓ
ｌＹ（￡）一ｇ（）Ｉ ≤３
１引言和定义
函数方程稳定性起源于１９４０年．［１从那时开始，很多数学家研究了各种各样的函数方程的稳定
性．因此研究微分方程的Ｈｙｐｅｒｓ．Ｕｌａｍ稳定性是很
第３４卷第５期

酰胺树枝状热可逆凝胶Supramolecular Ordering of Amide

Supramolecular Ordering of Amide Dendrons in Lyotropic andThermotropic ConditionsHye-Jin Jeon,†Min Kwan Kang,†Chiyoung Park,‡Kyoung Taek Kim,‡Ji Young Chang,§Chulhee Kim,*,‡and Hyun Hoon Song*,†Department of Ad V anced Materials,Hannam Uni V ersity,Daejeon,Korea,Department of Polymer Science and Engineering,Inha Uni V ersity,Incheon,Korea,and College of Materials Science and Engineering,Seoul National Uni V ersity,Seoul,KoreaRecei V ed August1,2007.In Final Form:September25,2007Self-assembled superstructures of amide dendrons,from first to third generation including monodendrons and covalently linked dimers,were extensively examined,and the supramolecular ordering processes in thermotropic and lyotropic conditions were compared.The superstructures as determined by X-ray diffraction and DSC revealed that the first and second generation dendrons showed nearly identical superstructures regardless of the assembly conditions. But,the third generation dendrons showed a more sensitive self-organizing behavior.The structure obtained from the gel state was lamellar with a more extended conformation,while the structure from the melt state revealed the columnar superstructures of contracted branches.The superstructure formed from the gel state also showed a structural change upon raising the temperature and assumed a structure similar to the thermotropically driven one,implying that the structure formed from the gel is thermodynamically unstable.The formation of lamellar-or cylinder-type superstructures from amide dendrons was primarily dependent on the shape of dendrons,which is associated with the branch size (generation)and the surrounding conditions.IntroductionA material having an ordered structure in the nanometer scale provides unique physical and chemical properties.Among many techniques to fabricate nanostructured materials,one approach is based on the self-assembling nature of large molecules such as linear polymers or dendrimers.In particular,perfect mono-disperse dendrimers and their self-assembling process have emerged as one of the main research area in such efforts.1-7 Self-assembly of dendron building blocks via secondary interactions such as hydrogen bonding,ionic interactions, hydrophilic properties,π-πinteractions,and metal-ligand coordination has been reported by several researchers.8-17It has been demonstrated that the self-assembly of dendrons is a unique and useful method to create well-defined superstructures at the nanometer scale.These researchers also reported that the self-assembly of dendrons results in different superstructures depend-ing on the assembly conditions including generation,molecular structures,solvent(or solubility),and temperature.18-23For example,Percec and co-workers published the self-assembly in which flat tapered and conical monodendrons generate cylindrical or spherical supramolecular structures depending on the size and shape of the monodendrons.9-11Self-organization behavior of some dendritic bolaamphiphiles in the aqueous phase was also elucidated by Newkome and co-workers.13,14Newkome reported that the introduction of rigidity or interactive internal functionality plays a critical role in the aggregation process for stacking an orthogonal or ordered helical array.24Variation of superstructure formation via self-assembly is not restricted in dendrimers.Variations have been reported in phase transitions of block copolymers including rod-coil block copolymers and the self-assembly of amphiphilic materials.25-30*To whom correspondence should be addressed.E-mail:(C.K.) chk@inha.ac.kr or(H.H.S.)songhh@hnu.kr.†Hannam University.‡Inha University.§Seoul National University.(1)Fre´chet,J.M.J.Proc.Natl.Acad.Sci.U.S.A.2002,99,4787.(2)Zimmerman,S.C.;Lawless,L.J.Top.Curr.Chem.2001,217,95.(3)Emrick,T.;Fre´chet,J.M.J.Curr.Opin.Colloid Interface Sci.1999,4, 15.(4)Genderen,M.H.;Meijer,E.W.Supramolecular Materials and Technolo-gies;Wiley:New York,1999;pp47-88.(5)Newkome,G.R.;Moorefield,C.N.Chem.Re V.1999,99,1689.(6)Zeng,F.;Zimmerman,S.C.Chem.Re V.1997,97,1681.(7)Hecht,S.;Fre´chet,J.M.J.Angew.Chem.,Int.Ed.2001,40,74.(8)Percec,V.;Glodde,M.;Bera,T.K.;Miura,Y.;Shiyanovskaya,I.;Singer, K.D.;Balagurusamy,V.S.K.;Heiney,P.A.;Schnell,I.;Rapp,A.;Spiess,H.-W.;Hudson,S.D.;Duan,H.Nature(London,U.K.)2002,417,384.(9)Hudson,S.D.;Jung,H.-T.;Percec,V.;Cho,W.-D.;Johansson,G.;Ungar,G.;Balagurusamy,V.S.K.Science(Washington,DC,U.S.)1997,278,449.(10)Percec,V.;Johansson,G.;Ungar,G.;Zhou,J.J.Am.Chem.Soc.1996, 118,9855.(11)Balagurusamy,V.S.K.;Ungar,G.;Percec,V.;Johansson,G.J.Am. Chem.Soc.1997,119,1539.(12)Percec,V.;Cho,W.-D.;Mo¨ller,M.;Prokhorova,S.A.;Ungar,G.; Yeardley,D.J.P.J.Am.Chem.Soc.2000,122,4249.(13)Newkome,G.R.;Baker,G.R.;Saunders,M.J.;Russo,P.S.;Gupta,V. K.;Yao,Z.;Miller,J.E.;Bouilion,K.J.Chem.Soc.,mun.1986,752.(14)Newkome,G.R.;Moorfield,C.N.;Baker,G.R.;Behera,R.K.;Escamillia,G.H.;Saunders,M.J.Angew.Chem.,Int.Ed.Engl.1992,31,917.(15)Zimmerman,S.C.;Zeng,F.;Reichert,D.E.C.;Kolotuchin,S.V.Science (Washington,DC,U.S.)1996,271,1095.(16)Ungar,G.;Liu,Y.;Zeng,X.;Percec,V.;Cho,W.-D.Science(Washington, DC,U.S.)2003,299,1208.(17)Zubarev,E.R.;Parlle,M.U.;Sone,E.D.;Stupp,S.I.J.Am.Chem.Soc. 2001,123,4105.(18)Isaacs,L.;Chin,D.N.;Bowden,N.;Xia,Y.;Whitesides,G.M.;Reinhoudt,D.N.Supramolecular Materials and Technologies;Wiley:New York,1999;pp 1-23.(19)Percec,V.;Cho,W.-D.;Ungar,G.;Yeardley,D.J.P.J.Am.Chem.Soc. 2001,123,1301.(20)Percec,V.;Cho,W.-D.;Mosier,P.E.;Ungar,G.;Yeardley,D.J.P.J. Am.Chem.Soc.1998,120,11061.(21)Percec,V.;Cho,W.-D.;Ungar,G.;Yeardley,D.J.P.Angew.Chem.,Int. Ed.2000,39,1597.(22)Percec,V.;Rudick,J.G.;Peterca,M.;Wagner,M.;Obata,M.;Mitchell,C.M.;Cho,W.-D.;Balagurusamy,V.S.K.;Heiney,P.A.J.Am.Chem.Soc. 2005,127,15257.(23)Peterca,M.;Percec,V.;Dulcey,A.E.;Nummelin,S.;Korey,S.;Ilies, M.;Heiney,P.A.J.Am.Chem.Soc.2006,128,6713.(24)Newkome,G.R.;Lin,X.;Yaxiong,C.;Escamilla,.Chem. 1993,58,3123.(25)Lee,M.;Cho,B.-K.;Ihn,K.J.;Lee,W.-K.;Oh,N.-K.;Zin,W.-C.J.Am. Chem.Soc.2001,123,4647.(26)Vaidya,N.Y.;Han,C.D.;Kim,D.;Sakamoto,N.;Hashimoto,T. Macromolecules2001,34,222.(27)Lee,S.-H.;Char,K.;Kim,G.Macromolecules2000,33,7072.13109Langmuir2007,23,13109-1311610.1021/la7023555CCC:$37.00©2007American Chemical SocietyPublished on Web11/16/2007Important factors to form various nanostructure are molecular size and shape,fraction or ratio between blocks,nature of solvents,concentration,temperature,etc.In our recent works,we also reported that amide based dendritic molecules form superstructures via self-assembly in various conditions.31-34The key structural elements in the self-assembly of the amide dendron were amide branches for hydrogen bonding,carboxyl functionality at the focal point,and alkyl tails for the stabilization of assembled structures via van der Waals interac-tions.Our results demonstrated that the amide dendrons formthermoreversible gels through self-organization in organic media and result in the formation of lamella or columnar superstructures in the dry state depending on the structures of the dendron building blocks.31,32It was also claimed that formation of the dimer units either by the hydrogen bond at the carboxylic focal point or by the covalent linkage at the focal point was the key structural prerequisite for self-assembly into superstructures.In this work,we carefully examined the effects of temperature variation on the self-assembly and assembled structures of amide(28)Sakamoto,N.;Hashimoto,T.Macromolecules 1998,31,8493.(29)Kimishima,K.;Koga,T.;Hashimoto,T.Macromolecules 2000,33,968.(30)Hanley,K.J.;Lodge,T.P.;Huang,C.-I.Macromolecules 2000,33,5918.(31)Kim,C.;Kim,K.T.;Chang,Y.;Song,H.H.;Cho,T.-Y.;Jeon,H.-J.J.Am.Chem.Soc.2001,123,5586.(32)Kim,C.;Lee,S.J.;Lee,I.H.;Kim,K.T.;Chang,Y.;Song,H.H.;Jeon,H.-J.Chem.Mater.2003,15,3638.(33)Ko,H.S.;Park,C.;Lee,S.M.;Song,H.H.;Kim,C.Chem.Mater .2004,16,3872.(34)Park,C.;Choi,K.S.;Song,Y.;Jeon,H.-J.;Song,H.H.;Chang,J.Y.;Kim,ngmuir 2006,22,3812.Figure 1.Chemical structures of amide dendrons (n G,no.of generations and D,dimeric dendrons).13110Langmuir,Vol.23,No.26,2007Jeon et al.dendrons.The results were then compared with those of lyotropic conditions in organic media.In particular,for the detailed structural investigation associated with self-assembly,high resolution and time-resolved X-ray scattering was performed utilizing the intense synchrotron radiation source.DSC was also utilized to provide thermal responses of the self-assembled superstructures.Experimental ProceduresMaterials.The chemical structures of dendrons studied in this work are depicted in Figure 1.Each dendron consists of a carboxylic group at the focal point,amide branches in the middle,and alkyl tails.Synthesis of the dendrons has been described elsewhere.31,35The solvent utilized for the self-assembly of dendrons into the gel state was chloroform for the first and second generation dendrons and dichloromethane for the third generation ones.Characterization.DSC (TA 2910)was used for the examination of thermal behavior of superstructures derived either from the gel state or from the melt state.For the DSC measurement,samples of ∼3mg were heated at 3°C/min under a nitrogen atmosphere.High resolution and time-resolved X-ray scattering was carried out at the Pohang Accelerator Laboratory and was operated at 2.0GeV and 120mA.The scattering intensities were measured using the diffractometer at the 3C2beamline,but for the in situ scattering,a position sensitive area detector was utilized at the 4C2beamline.Aluminum foil (12µm thick)was used for the window material of the sample holder instead of using the ordinary Kapton film.This was done to eliminate the scattering hump arising from the Kapton film,which overlaps with the scattering peak of the dendrons.For X-ray scattering at an elevated temperature,the sample was heated to a predetermined temperature based on the DSC thermogram,and the temperature was maintained for 5min to achieve temperature equilibration before the measurement.The scattering intensities were measured at diffraction angles of 0.4-23°with 0.05and 0.02°intervals that cover both small angles and wide angles simultaneously.The exposure time was 1s at each interval.Results and DiscussionSuperstructures of Amide Dendrons from Gel State.In Figure 2,X-ray diffraction patterns of dry gels taken at an ambient temperature are plotted.The plots reveal that the dendrons of first generation (1G and 1GD)and third generation (3G and 3GD)form lamellar superstructures,while those of second generation (2G and 2GD)form cylindrical columns of hexagonal packing.Here,1G stands for first generation monodendron and 1GD for the first generation dimer.The results of first generationand second generation are consistent with our previous reports.31The lamellar spacings and cylinder diameters derived from the main diffraction peak are summarized in Table 1.The mono-dendrons show somewhat higher packing order than the dimeric units.However,the monodendrons and dimers share the same superstructures with similar sizes,confirming that the dimer units linked either by covalent bond or by hydrogen bond at the focal carboxylic group are the basic building blocks for self-assembly as has been reported earlier.31-34The results also suggest that the formation of symmetrical unit is critically required for the molecules to organize into ordered superstructures.The formation of self-assembled superstructures is associated with the structural characteristics of the dendrons and their interactions with the surrounding molecules.As already discussed in the previous section,the driving forces of self-assembly of these amide dendrons are intermolecular hydrogen bonds between the amide groups and at the focal carboxylic group as well as the van der Waals interactions of the alkyl tails.For the first generation (1G and 1GD),the dendrons are relatively small with two amide branches.The steric hindrance is,therefore,minimal,and the low viscosity allows for easy packing.As can be noted in Table 1,the first generation dendrons form the lamellar structure with a nearly extended conformation (l /l 0)∼0.9).The X-ray patterns of first generation dendrons also exhibit many reflections at high angles,demonstrating that the crystalline order exists within the lamellar domains.The dendrons of the third generation (3G and 3GD)also show a lamellar structure but with a much less ordered feature.The third generation dendrons are large and bulky,and the overall molecular shape becomes globular.The value l /l o )∼0.7shown in Table 1suggests the contracted form of the dendrons in the superstructure formation.Formation of Lamellar-or Cylinder-Type Superstructures.We have seen that first and third generation dendrons form the lamellar-type superstructure,while the second generation den-drons assemble into columnar cylinders.As was emphasized previously,the formation of the dimer unit is an essential process for the self-assembly of monodendrons into an ordered super-structure.The chemical structures of the dendrons depicted in Figure 1reveal that the dimer unit takes a dumbbell shape,where the central linkage core has a relatively low density.We can envisage that this causes some frustration in the packing of the molecules side by side,which becomes more distinct with the increase of generation.For the dimers to pack efficiently,dimer units must cross-pack one another.The cross-packed units must stack side-by-side along the long axis to yield a lamellar structure.On the other hand,for cylindrical column formation,each dumbbell-shaped dimer unit must cross-pack continuously along the long (cylinder)axis,resulting in a spiral structure and thus a cylindrical superstructure.The assembly processes are sche-matically described in Figure 3.With regard to forming the lamellar or cylindrical super-structure,it is worth mentioning that the gelation time of the third generation is rather shorter than the second generation dendrons.A high number of branches and thus hydrogen bonding sites may promote the gelation process and the assembly process.The second generation,on the other hand,taking the molecular shape between the first and the third generation,shows the slowest gelation time among the three generations.The number of branches and,therefore,the hydrogen bonding sites are higher than those of the first generation but may not be high enough to compensate for the steric hindrance arising from the bulky size.The size and shape of the branch units and their interactions within the surrounding solvents are all important parameters in determining the ordering process.(35)See SupportingInformation.Figure 2.X-ray diffraction patterns of dendrons in the dry gel state at ambient temperature.Supramolecular Ordering of Amide Dendrons Langmuir,Vol.23,No.26,200713111Thermotropically Induced Superstructures of Amide Dendrons.The dry gels of dendrons having a supramolecular order were raised above the disordering temperature,and a new thermotropically driven structure was obtained upon cooling the dendrons.X-ray scattering patterns of the dendrons taken at ambient temperature are plotted in Figure4.The first generation dendrons,1G and1GD,show similar lamellar patterns to those of the dry gel.When compared with the patterns of the dry gel, the patterns of the first generation dendrons exhibit a broad main peak and a lack of diffraction peaks at high angles,illustrating the reduced packing order both in superstructure and in local structures.In particular,we also further note an extra peak at the very small angles of the1G pattern.This interesting structure will be discussed in the following section.For the second and third generations,on the other hand,both the monodendron(2G and3G)and the dimer(2GD and3GD)show a hexagonally packed supramolecular order but with a much broader main peak and absence of high order peaks.It is also noted that the dendrons of thermotropically driven structures are more contracted than those of the gel state,which is more distinctive for the third generation dendrons as indicated in Table1.The reduced packing order and contracted conformation observed in these thermo-tropically driven structures are associated with the sluggish motion of the dendrons in the crowded melt state.We also emphasize that the third generation dendrons(3G and3GD)show the superstructure of hexagonally packed cylinders(we recall that the dry gel has shown the lamellar structure).It is very interesting to observe that the third generation dendrons can form either the lamellar structure or the cylinders depending on the assembly conditions.As discussed in the previous section,for the cylindrical superstructure to be formed,the dimeric dendrons must stack with continuous rotation along the cylinder axis.The cylinder-type superstructure appears to be more favored for the contracted dendrons of higher steric hindrance in this crowded environment. Thermal Effects on Superstructures.Now,we examine the structural change induced upon elevating the temperature for both samples derived from the gel state of the lyotropic environment and from the melt of the thermotropic environment. The former is called the first heating and the latter the second heating,hereafter.X-ray scattering patterns were measured at the predetermined temperatures upon raising the temperature, and the thermal data of DSC in Figure5were used as a guideline. Before measuring the X-ray pattern,the predetermined tem-perature was brought to an equilibrium by maintaining the sample for5min at each temperature.In Figure6(1G-1),X-ray patterns of1G at various temperatures are plotted,while the corresponding DSC thermogram is shown in Figure5(1G).The initial lamellar structure is maintained without any marked change until it melts into a disordered liquid. We also note that the primary peak of the superstructure andTable1.Molecular Dimensions of Dendrons and Interdomain Distances of Self-Assembled Superstructures1G1GD2G2GD3G3GDsuperstructure a L L C C L Lmolecular dimension b l0(Å)265336734591 Interdomain distance c l g(Å)454757596567l g/l00.87d0.890.79d0.810.72d0.74l m(Å)4652535253l m/l00.870.710.730.570.58a L,lamella and C,cylinder.b Stretched molecular dimension(l0)of the dendrons was obtained by using a Cerius2on a Silicon Graphics O2 workstation.c Interdomain distances(lamellar thickness or cylinder diameter)were derived from the X-ray results;l g denotes the dimension ofgel-state-driven structure and l m for the melt-induced one.d2l0was used for the calculation of monodendrons(1G,2G,and3G).Figure3.Schematic representation of supramolecular ordering of amide dendrons into lamellae or cylindricalcolumns.Figure4.X-ray diffraction patterns of thermally induced super-structures of amide dendrons at ambient temperature.13112Langmuir,Vol.23,No.26,2007Jeon et al.those of the local order within the superstructure domain disappears nearly simultaneously at temperatures of 109-128°C.But,in the second heating shown in Figure 6(1G-2),somewhat different patterns are revealed.Before discussing the temperature effect,we begin with the X-ray patterns of the melt grown sample at room temperature.The X-ray pattern of 1G obtained at room temperature (Figure 6(1G-2))reveals the first two peaks located at s )0.0144and 0.0283Å-1,which correspond to 70and 36Å,respectively.The first peak intensity is much lower than the second peak,and the 70Åspacing is far beyond the extended length of the dimer unit.It is apparent that the two peaks do not arise from single structural units,in this case the lamellae.The pattern implies that the thermotropically driven structure consists of lamellae of two different layer thicknesses.The population of the lamellae with larger thicknesses is relatively minor.As the temperature is raised,however,the two peaks merge into one that represents a well-ordered lamellar structure with spacing of 41.3Å.The structure is very similar to that of the dry gel (Figure 2).When increasing the temperature further,the spacing increases to 44.6Åwith the advent of higher order peaks,illustrating an improvement of the supramolecular packing order.It is very interesting to observe that two different lamellar units of 70and 36Åform when the structure is derived from the disordered melt.We recall that forming the dimer unit is an essential process for the supramolecular ordering and also that the extended molecular length of the dimer unit of 1G is 53Å(Table 1).In a dilute solution,hydrogen bonding between the carboxylic units at the focal point appears to be preferred over other possible hydrogen bondings between carboxylic group and amide branches,yielding a primary building block of the dimer unit as demonstrated by the structures formed from solution.But,in a melt of a molecularly crowded situation,the structural rearrangement is rather restricted and may not be fast enough to follow the temperature change,allowing hydrogen bonding between carboxylic units and amide branches.In Figure 7a,a possible packing mode to result in the lamellar structure of two different layer spacings (70and 36Å)is depicted.The severely contracted alkyl branches are expected in these two lamellar units as illustrated in the figure.When sufficient thermal energy is supplied at an elevated temperature,the structure,however,restores itself to the stable one (Figure 7b)that is linked by hydrogen bonding between the carboxylic groups at the focal point and with more extended alkyl branches.The exothermic hump at 50-60°C in the heating curve of the DSC thermogram (Figure 5(1G-2))can be associated with the structural rear-rangement to form a new lamellar structure of single layer thickness.The X-ray patterns of 1GD (first generation dimer)are depicted in Figure 6(1GD-1and 1GD-2).The pattern 1GD-1from the gel state reveals that the structure is not as orderedasFigure 5.DSC thermograms of dendrons;curve 1denotes the first heating (heating of dendrons derived from the gel state)and curve 2the second heating (heating of thermally induced dendrons,exo up).Supramolecular Ordering of Amide Dendrons Langmuir,Vol.23,No.26,200713113Figure 6.X-ray diffraction patterns obtained with temperature variation;nG -1denotes the first heating and nG -2for the second heating.13114Langmuir,Vol.23,No.26,2007Jeon et al.that of 1G as can be expected as the dimer units covalently linked at the focal point are relatively restricted in the movement of the assembly process.However,the pattern 1GD-2(1GD from the melt state)shows a somewhat more ordered packing state than that from the gel state (1GD-1),which is contradictory to the results of 1G.Also,in this case,a single lamellar spacing is observed throughout the whole temperature range.It is apparent that the covalently linked dimer units are no longer able to form the structure suggested in Figure 7a and that they are able to form better packing than 1G before the structural reorganization takes place at an elevated temperature.DSC results of second generation dendrons,depicted in Figure 5(2G and 2GD),show more complicated thermal behavior thanthose of the first generation.Corresponding X-ray patterns of second generation dendrons (2G and 2GD)upon heating are denoted as 2G-1,2G-2,2GD-1,and 2GD-2in Figure 6.Results of second generation dimers (2GD-1and 2GD-2)are very similar to those of monodendrons (2G-1and 2G-2).All patterns exhibit the hexagonally packed cylinder-type supramolecular order without distinctive high angle peaks.It is certain that the local order within the supramolecular domain is not required for the formation of the superstructure.Careful comparison between the patterns,however,reveals that the structure of monodendron from the gel state (2G-1)shows a somewhat better packing order than thermotropically driven structures and also those of dimeric dendrons.We also note that the main peak sharpens with the temperature increase,illustrating that the supramolecular order is improving.The ordering effect is more distinct in the heating of the melt quenched samples (second heating).The exotherms observed in the DSC thermograms are apparently associated with the ordering of the structures.When the temperature was raised further above the final endotherms in DSC,the main peak at small angles disappeared,leaving a new broad hump representing the disordered structure.X-ray patterns of third generation dendrons,3G and 3GD,are also plotted in Figure 6.Here,3G-1and 3GD-1denote the dry gels obtained from the solution,while 3G-2and 3GD-2denote the samples from the melt state.The patterns of the dry gels (3G-1and 3GD-1)exhibit a lamellar structure,while the structure from the melt state (3G-2and 3GD-2)confirms the hexagonal cylinders as already discussed in the previous section.The initially formed supramolecular order becomes improved upon heating until the DSC endotherm appears as was observed with the previous first and second generation dendrons.One particular observation,however,noted in the heating of the dry gel is that a transition of the superstructure appears to take place at the elevated temperatures during the heating process.For detailed examination,time-resolved X-ray scattering using asynchrotronFigure 7.Schematic representation of thermotropically induced superstructure of 1G showing dual lamellar thickness (a)and rearranged structure at an elevated temperature (b).Note that dendrons on right in panel a are arranged to emphasize hydrogen bonds between the focal carboxyl groups and the amidebranches.Figure 8.Structural change of third generation dendrons with temperature variation:3G (a)and 3GD(b).Figure 9.X-ray diffraction patterns of dendrons above the disordering temperature:(a)1G,2G,and 3G and (b)1GD,2GD,and 3GD.Supramolecular Ordering of Amide Dendrons Langmuir,Vol.23,No.26,200713115radiation source and a position sensitive area detector was performed upon heating the samples (3°C/min).The results of 3G and 3GD are depicted in Figure 8a,b,respectively.In both figures,we indeed note a drastic change of the diffraction peaks at temperatures between 60and 90°C,which is apparently responsible for the first endothermic peak of DSC thermograms shown in Figure 5(3G-1and 3GD-1).The main peak representing the lamellar structure of ∼70Åthickness rapidly shifts to a higher angle,which also represents another lamellar structure but with a much reduced thickness of ∼48Å.Therefore,the changes noted in the diffraction pattern are associated with the internal rearrangement within the super domains while maintain-ing the original skeleton.The thickness of the second lamella (48Å)is extraordinarily small,considering the initial lamellar thickness (∼70Å)and the molecular dimension of the third generation dendrons (91Å),implying that the branches of the dendrons are severely contracted.The dendrons assembled into the superstructure from the solution are in a more extended conformation,which seems to be thermodynamically unstable.Once the thermal energy is available at an elevated temperature,the branches undergo rearrangement to a more stable contracted conformation.At this point,we recall that the structure formed from the melt state was the hexagonally packed cylinder,where the cylinder diameter was 55Å.The lamellar structure with contracted branches observed during the heating might be the intermediate stage of transition from the lamellar to the equilibrium cylindrical domains.Disordered Melt State.As discussed in the previous section,all dendrons showed a transition from the ordered supramolecular structure to a disordered state at temperatures above the final melting endotherms,seen in the DSC thermal scans (Figure 5).It is noted that the melting endotherms of dimers are located at higher temperatures than those of monodendrons.The result isconsistent with all generations and can be attributed to the low entropy gain in the melting of the superstructures of the dimer units.At the disordered melt state,the molecules are randomly dispersed with a uniform density distribution throughout the entire sample.However,in X-ray scattering patterns taken from the melt state shown in Figure 9,we note a broad but strong hump at the small angles (∼30Å),implying the existence of a structure possessing a somewhat strong correlation even in the melt state.The pattern with a strong correlation of ∼30Åalso suggests that each dendron molecule isolates itself from the others and forms an individual particle.The correlation distance and fwhm estimated from the hump are shown in Figure 10.The correlation distance increases with the generation from 26to 33Å.The fwhm of the peak,on the other hand,decreases with the increase of generation,indicating the higher coherency in the high generation dendrons.Dimers apparently show the higher coher-ence over the monodendrons,which is more marked in the first generation dendron.As already was discussed,the hydrogen bonds at the focal point and between the amide branches are the main driving forces to form the superstructure.The transition of the ordered superstructure to a disordered melt at high temperature must be accompanied by the breakage of these hydrogen bonds.It is,however,interesting to note that both monodendrons and covalently linked dimers show nearly the same correlation distances in the melt state.Even in the covalently linked dimers (1GD,2GD,and 3GD),the dendronic branches in either side of the focal linkage behave like the individual monodendrons.ConclusionDendronic molecules containing amide branches including monodendrons and covalently linked dimers (up to the third generation)were studied for their supramoleular ordering in both lyotropic and thermotropic environments.The supramolecular ordering of these dendrons differs depending on the assembling conditions as well as the generations.The differences associated with the two different passages are more distinct with the higher generation dendrons.The dendrons assembled in the lyotropic conditions showed a better supramolecular ordering,but the dendrons forming the superstructure appear to be thermody-namically unstable.When sufficient thermal energy is supplied,the dendrons relax to the more stable state,which is similar to the behavior of those in thermotropic conditions.The formation of symmetric molecular entities is the key process for monodendrons to self-organize into superstructures.As has been claimed previously,the dimer units formed either by hydrogen bonding between the carboxylic units of mono-dendrons or the covalently linked units at the focal points are the basic building blocks for ordered superstructures.The dimer units of amide dendrons,however,result in an unfavorable shape for efficient packing.The dimer units must cross-pack to one another to avoid steric hindrance of the bulky branches.The cross-packed dendrons stack to result in either lamellar or columnar superstructures.Acknowledgment.This work was supported by Korea Research Foundation grants (MOEHRD)(KRF-2003-202-D00256)and by Korea Science and Engineering Foundation grants (MOST)(R01-2005-000-10343-0)funded by the Korean government.The X-ray scattering was performed at the 3C2,4C1,and 4C2beamlines at the Pohang Accelerator Laboratory.Supporting Information Available:Synthesis of third genera-tion dendrons (3G and 3GD).This information is available free of charge via the Internet at .LA7023555Figure 10.Correlation distance and fwhm estimated from X-ray patterns of the disordered melt.13116Langmuir,Vol.23,No.26,2007Jeon et al.。

亥姆霍兹方程

热力学定律
一切过程都必须遵循，保持能量守恒；不能解决过程是否必然发生、进行的程度
热力学第二定律——判断在指定的条件下一个过程能否发生；如能发生的话，能进行到什么程度；如何改变外界条件（温度、压力等）才能使变化朝人们所需要的方向进行
东莞理工学院
Dongguan University of Technology
东莞理工学院
Dongguan University of Technology
广东省分布式能源系统重点实验室
3.1.2 热力学第二定律
热力学第二定律：在不违背热力学第一定律的前提下，判断在一定条件下过程的方向和限度的定律
“自发过程都是热力学不可逆过程”这个结论是人类经验的总结，也是热力学第二定律的基础
东莞理工学院
Dongguan University of Technology
广东省分布式能源系统重点实验室
3.3.1.3 p、V、T都改变的过程
东莞理工学院
Dongguan University of Technology
广东省分布式能源系统重点实验室
东莞理工学院
Dongguan University of Technology
广东省分布式能源系统重点实验室
熵变计算的基本公式
当始、终态一定时，不论过程是否可逆，其熵变都可用下式求出：
不论过程是否可逆，都必须通过可逆过程的热温商来计算熵变；如果过程是不可逆的，应设计一个与该不可逆过程的始、终态相同的可逆过程
东莞理工学院
Dongguan University of Technology
自然界的自发过程多种多样，但人们发现自发过程都是相互关联的，从某一个自发过程的不可逆性可以推断另一个自发过程的不可逆性。因此热力学第二定律的表述也有多种，但它们都是等价的

2.选择性提取冶金原理

2.1.2铁液元素的 rG －T
1）[Cu]、[Ni]、[Mo]、[W]等氧化的线均在[Fe]线之上。 2 ） [P] 的线也在 [Fe] 线之上，吹氧不会使 [P] 氧化，铁水中的 [P] 只能通过造碱性渣去除一部分。 3 ） [Si] 、 [B] 、 [Ti] 、 [Al] 、 [Ce] 的线均在 [Fe] 线之下，吹氧时最容易氧化，它们是强脱氧剂。 4） [Cr]、[Mn]、[Nb]等元素的线与 [C] 氧化的相交，因此存在氧化转化温度。温度高于 T转时 [C]氧化，低于 T转时元素氧化。
r G (465260 307.69T ) J mol1
2 1/ 2 f [ C ] a Cr3O4 C r G r G RT ln 3 / 2 f Cr [Cr ]3 / 2 ( pCO / p ) 2
2.1.3 利用 T转分析不锈钢冶炼过程中的“去碳保铬”
2.303RThpH(1)
上式即为E—pH关系式，其中R=8.314J·K-1·mol-1； F=96500C·mol-1。若将T=298K代入上式：
b a nEF r G 5705 .85lg B 5705 .85hpH r (2) aR
2.3.1 E-pH图绘制原理
2.2氧化物选择性还原平衡图
在沸腾炉内用 CO作为还原剂还原矿石，使得矿石中得 Ni 、 Co 被还原进入金属溶液，而铁则不被还原，以氧化物的形式留在矿石中，从而达到去除 Ni 、 Co 的目的。铁为变价金属，被CO逐级还原：
3Fe2O3 (s) CO 2Fe3 O4 (s) CO2 r G (26520 57.03T ) J mol1
r G r G RT ln

函数方程和微分方程的Hyers-Ulam稳定性的开题报告

函数方程和微分方程的Hyers-Ulam稳定性的开题报告1. 研究背景和意义：函数方程和微分方程是数学中的重要分支，它们在物理、化学、工程等领域中有广泛的应用。

研究这些方程的稳定性是解决一些实际问题的基础，也是理论问题的重要部分。

Hyers-Ulam稳定性是函数方程和微分方程中的一个重要概念，它描述了微小变化对方程解的影响程度，具有重要的实际应用价值。

2. 研究内容和方法：本论文将主要研究函数方程和微分方程的Hyers-Ulam稳定性，包括如下两个方面：（1）函数方程：研究给定函数方程的解在输入微小扰动后的变化情况，探讨Hyers-Ulam稳定性和S和R性质之间的关系。

常见的函数方程有加法方程、多项式方程、函数方程组等，我们将综合运用代数、分析等数学方法对这些方程进行研究。

（2）微分方程：研究微分方程的解在初始条件微小扰动下的稳定性，构造有限时间的Lyapunov函数刻画方程解的稳定性，在此基础上研究方程的全局稳定性。

我们将综合应用微分方程、稳定性理论等数学工具，尤其是利用Lyapunov函数等方法进行分析和证明。

3. 预期结果：研究器函数方程和微分方程的Hyers-Ulam稳定性，通过证明一些定理和推导一些条件，获得如下预期结果：（1）探究Hyers-Ulam稳定性和S和R性质之间的关系，构建一些新的例子和应用。

（2）证明一些函数方程的解在输入微小扰动后的稳定性定理，结合具体问题实例解释应用价值。

（3）研究微分方程的稳定性定理，探讨全局稳定性的充分条件，深入挖掘解的演化规律。

（4）对所获得的结果进行统计、分析和总结，提出一些实际问题的应用建议，并指出未来研究的方向和问题。

4. 研究创新点：（1）综合应用代数、分析、微分方程等多学科的理论和方法，创新性地研究函数方程和微分方程的稳定性。

（2）将Hyers-Ulam稳定性和S和R性质联系起来研究，扩展了其中一个方向的研究。

（3）构造Lyapunov函数分析微分方程的稳定性，从一个新的角度深入探究微分方程的全局稳定性。

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Jensen型方程的Hyers-Ulam稳定性
作者：学位授予单位：张冬燕福建师范大学
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福建师范大学硕士学位论文 Jensen型方程的Hyers-Ulam稳定性姓名：张冬燕申请学位级别：硕士专业：基础数学指导教师：王建 20070401
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