Jensen型方程的HyersUlam稳定性

合集下载

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
３若）
’

拟Banach代数上拟同态的Hyers-Ulam—Rassias稳定性(Ⅰ)

究．－ｉｃｏｘｓｎ．ｄ．ｎＥｍａ：ａｈ￣ｎｕｅｕｃｌ
４２２
纺
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。

线性微分方程y″-λ 2y=f(x)的Hyers-Ulam稳定性

明存在与ｅ和无关的常数志使得ｌ一ｌ，＜ｅ某个满足 “ 。一厂ｚ对一 “ （）的某个二阶连续可微函数 “
成立．
— — 收稿日期：２０１００８０２
基金项目：国家自然科学基金资助项目（０７２３１８１１）
Ｖｏ．０ＮＯ１４．１
Ｊｎ２１ａ．０１
线性微分方程一２（Ｙ一ｚ）的Ｈｙｒ— ａ稳定性ｅ州松田职业学院，东广州５１７；２１广广１３０．中山大学数学系，东广州５０７）广１２５
Ｉ一。一－ｚｌｅ则存在 “∈ Ｃ（）ｙ厂）＜，（。Ｊ满足Ｕ — “ （），ｔ一／ｚ一０并且ｌ（）（）＜是．ｚ一“ｚＩｅ
１主要结果
定理若二阶连续可微函数ｙ卜＋尺，：满足微分不等式
ｌ一。＿ｚｆｅｙ一厂） ≤ ，（
关键词：ｅｓＵａ稳定性；微分方程；函数方程Ｈｙｒ— １ｍ
中图分类号：０７．１７２文献标志码：Ａ文章编号：１０－７５２１）１０１－３０１８３（０１０－０８０
０引言
１４９０年，．ＵｌｍＩ提出关于函数方程稳定性的问题，问题关于可加接近映射在ＢｎｃＳＭ．ａ１该ａａｈ空间的情形已由ＤＨ．ｐｒ＿解决，被Ｔ．Ｒｓｉｓ］广．．Ｈｙｅｓ２并ｈＭ．ａｓ【推ａ３此后，多的学者口深入研究了函数方程的稳定很

Cauchy-Jensen方程的模糊稳定性

文献标志码：Ａ
１９４０年，Ｕｌａｍ［１提出了函数方程稳定性的问题，Ｈｙｅｒｓ在文献［２］中解决了近似可加映射的稳定性问题．１９７８年，Ｒａｓｓｉａｓ在文献１－３］中给出了推广，Ｇａｖｒｕｔａ也在文献［４］给出了Ｈｙｅｒｓ — Ｕｌａｍ— Ｒａｓｓｉａｓ定理的推
定义２映射厂：ｘ× Ｘ— ｙ称为二元可加映射，指厂关于每一个变元为Ｃａｕｃｈｙ函数。引理１设映射，：ｘ×ｘ— ｙ为Ｃａｕｃｈｙ — Ｊｅｎｓｅｎ映射，如果，（，０）一０，Ｖｚ∈ Ｘ，则，为二元可加映射，特
定理设Ｘ是实向量空间，Ｙ是实Ｂａｎａｃｈ，空间，映射：Ｘ一［０，ｃｏ），满足Ｖｚ，，，硼 ∈ Ｘ，都有
ｚ）＋ｆ（ｘ，锄）＋ｆ（Ｙ，ｚ）＋ｆ（Ｙ，叫）的稳定性。关键词：Ｃａｕｃｈｙ — Ｊｅｎｓｅｎ方程；稳定性；直接方法中图分类号：Ｏ１７７．１
主题分类号：３９Ｂ５２，４６ＨｌＯ
Ｃａｕｃｈｙ — Ｊｅｎｓｅｎ方程的模糊稳定性
卢海宁，纪培胜，薛海燕
（青岛大学数学科学学院，青岛２６６０７１）
摘要：用直接方法证明了Ｂａｎａｃｈ空间上的Ｃａｕｃｈｙ — Ｊｅｎｓｅｎ方程２厂（ｚ＋，）＝厂（ｚ，

分数阶微分方程的Hyers-Ulam稳定性

分数阶微分方程的Hyers-Ulam稳定性分数阶微分方程的Hyers-Ulam稳定性引言：在数学领域中，稳定性是一个重要的概念。

稳定性分为两种类型：局部稳定性和全局稳定性。

在过去的几十年中，研究人员通常关注线性微分方程的稳定性。

然而，随着对非线性微分方程的研究兴起，人们开始关注非线性微分方程的稳定性。

其中一种特殊情况是分数阶微分方程的稳定性问题。

分数阶微分方程是一类具有导数和积分的微分方程，被广泛应用于物理学、工程学、生物学等领域。

本文将探讨分数阶微分方程的Hyers-Ulam稳定性。

一、分数阶微分方程的定义分数阶微分方程是一类一般化的微分方程，其中指数阶的导数和积分可以是非整数。

它们可以用Caputo导数或Riemann-Liouville积分来表示。

分数阶微分方程的一般形式如下：\[D^\alpha_t y(t) = f(t,y(t)), \quad 0 < \alpha < 1, \quad t \in [a,b]\]其中，$D^\alpha_t y(t)$是Caputo导数，$f(t,y(t))$是已知函数。

二、Hyers-Ulam稳定性的定义Hyers-Ulam稳定性是一个数学定理，描述了函数方程的连续近似解存在的条件。

给定函数方程$F(x,y) = 0$，如果对于任意给定的$t \in [a,b]$，存在连续函数$\delta(t)$满足\[F(x,y) + \delta(t) = 0\]其中，$\delta(t)$是满足条件$\delta(0) = 0$的函数，则称函数方程在给定区间上是稳定的。

Hyers-Ulam稳定性在线性分数阶微分方程的研究中得到了广泛的应用。

三、分数阶微分方程的Hyers-Ulam稳定性对于分数阶微分方程，其Hyers-Ulam稳定性问题的解决方法相对复杂。

一种方法是通过构造包络函数来研究解的存在性。

另一种方法是通过使用变换和积分等技巧推导解的递推关系。

jensen函数方程

jensen函数方程（最新版）目录1.Jensen 函数方程的定义和基本概念2.Jensen 函数方程的性质和特点3.Jensen 函数方程在数学领域的应用4.Jensen 函数方程的求解方法和技巧5.总结正文一、Jensen 函数方程的定义和基本概念Jensen 函数方程，又称为 Jensen 方程，是由丹麦数学家 Jensen 提出的一类具有特殊性质的函数方程。

Jensen 函数方程广泛应用于数学的各个领域，如微积分、概率论、数理统计等。

它描述了在某种意义下，一个凸函数与其导数之间的关系。

二、Jensen 函数方程的性质和特点Jensen 函数方程具有以下几个基本性质：1.凸性：Jensen 函数是一个凸函数，这意味着对于其定义域内的任意两个实数，函数值总是介于这两个实数的函数值之间。

2.可微性：Jensen 函数在其定义域内具有连续的导数。

3.对称性：Jensen 函数方程对于自变量的任意置换都具有对称性。

4.Jensen 不等式：Jensen 函数的导数与函数本身之间存在一种特殊的不等关系，这就是著名的 Jensen 不等式。

三、Jensen 函数方程在数学领域的应用Jensen 函数方程在数学的许多领域都有广泛应用，如求解最优化问题、研究随机过程、分析概率分布等。

其中，最著名的应用之一是求解凸函数的最小值问题。

通过 Jensen 函数方程，我们可以将求解最小值问题转化为求解一个关于导数的方程，从而简化问题的求解过程。

四、Jensen 函数方程的求解方法和技巧求解 Jensen 函数方程的方法和技巧有很多，以下是一些常用的方法：1.利用 Jensen 不等式：根据 Jensen 不等式，我们可以得到一个关于函数导数的不等式，从而将原问题转化为求解一个不等式问题。

2.利用函数的凸性：由于 Jensen 函数是凸函数，我们可以利用函数的凸性来推导出一些有用的不等关系，从而简化问题的求解。

3.利用微分中值定理：微分中值定理是微积分中的一个基本定理，我们可以利用它来求解 Jensen 函数方程。

jensen函数方程 -回复

jensen函数方程-回复Jensen函数方程是一种关于函数的函数方程，它在数学和经济学等领域中有着广泛的应用。

该方程是由丹麦数学家Johan Ludwig Jensen在20世纪初提出的，它描述了一种特殊的类型函数的性质。

在这篇文章中，我们将深入探讨Jensen函数方程的定义、性质以及一些常见的解法。

首先，让我们来了解一下Jensen函数方程的定义。

给定一个定义在实数集上的函数f(x)，如果满足下面的条件：f((x+y)/2) = (f(x) + f(y))/2，对于任意的x和y成立那么函数f(x)就被称为满足Jensen函数方程。

简单来说，这个方程要求函数f在两点的中间值处取得的函数值等于这两点函数值的平均值。

接下来我们探讨一下Jensen函数方程的一些性质。

首先，我们可以发现对于线性函数来说，它肯定满足Jensen函数方程，因为对任意的x和y，有(f(x)+f(y))/2 = (ax+ay)/2 = a(x+y)/2 = f((x+y)/2)，其中a是任一常数。

这意味着所有的线性函数都是Jensen函数方程的解。

其次，Jensen函数方程还有一个重要的性质，即任意两点之间的任意插值都满足方程。

假设f(x)满足Jensen函数方程，那么对于任意的实数a1，a2和对应的函数值x1 = f(a1)和x2 = f(a2)，对于任意的0≤t≤1，有f(ta1+(1-t)a2) = tf(a1) + (1-t)f(a2)。

这个性质意味着我们可以通过线性插值的方式得到方程的解，而不局限于线性函数。

现在我们来看一些常见的Jensen函数方程的解法。

首先，我们可以通过验证函数的某些性质来判断它是否满足方程，例如函数的凸性和仿射性质。

对于凸函数来说，它始终在两点的中间值处取得的函数值小于这两点函数值的平均值，因此凸函数是Jensen函数方程的解。

而仿射函数则满足凸性的条件，并且对某一常数a有f(x) = ax + b形式，所以仿射函数也是方程的解。

算子方程的ε-Hyers-Ulam稳定性

算子方程的ε-Hyers-Ulam稳定性
许璐;曹怀信
【期刊名称】《纺织高校基础科学学报》
【年(卷),期】2008(021)001
【摘要】根据环同态的稳定性,引入了算子方程Ax=0的ε-Hyers-Ulam稳定性的概念.在此基础上,给出了算子方程Ax=0是ε-Hyers-Ulam稳定的一些充分必要条件.得到了(A)ε>0,算子方程Ax=0是ε-H-U稳定的当且仅当kerA是非空的.
【总页数】3页(P58-60)
【作者】许璐;曹怀信
【作者单位】陕西师范大学,数学与信息科学学院,陕西,西安,710062;陕西师范大学,数学与信息科学学院,陕西,西安,710062
【正文语种】中文
【中图分类】O177.1
【相关文献】
1.关于Jensen泛函方程Hyers-Ulam稳定性的一个结论 [J], 宋爱民
2.广义二次函数方程在模糊赋范空间上的Hyers-Ulam稳定性 [J], 崔哲男;王彪;赵顺实;李林松
3.微分方程的Hyers-Ulam稳定性 [J], 薛建明;张娟
4.二阶线性微分方程的Hyers-Ulam稳定性 [J], 薛建明
5.线性微分方程的Hyers-Ulam稳定性 [J], 杨月;孟凡伟
因版权原因，仅展示原文概要，查看原文内容请购买。

复习题-物理化学

第十二章化学动力学基础（二）【复习题】【1】简述碰撞理论和过渡态理论所用的模型、基本假设和忧缺点。

【解】碰撞理论模型：将反应物分子看成无内部结构刚性球体，它们的碰撞完全是弹性碰撞。

基本假设：是反应物分子只有经过碰撞才能发生反应，但并不是所有碰撞分子都能发生反应，只有当分子的相对碰撞能等于或超过临界能时才能发生反应。

优点：（1）碰撞理论为人们描述了一幅虽然粗造但十分明确的反应图象，在反应速率理论的发展中起了很大作用；（2）对Arrheinus公式中的指数项，指前因子或阈能提出了较明确的物理意义，认为指数项相当于有效碰撞分数，指前因子A相当于碰撞频率；（3）解释了一部分实验事实，理论所计算的速率常数k与较简单的反应实验值相符。

缺点：（1）要从碰撞理论来计算速率常数k，必须要知道临界能E c，它本身不能预言E c的大小，还需通过Arrheinus公式来求，而Arrheinus公式中的E a的求得，首先需要从实验测得k，这就使该理论失去了从理论上预言k的意义，说明该理论为半经验理论；（2）在该理论中曾假设反应物分子是无内部结构的刚性球体，这种假设过于粗糙，因此只对比较简单的反应，理论值与实验值符合的较好，但对更多的反应，计算值与实验值有很大的差别。

过渡态理论模型：描绘出势能面基本假设：a 化学反应不是通过简单的碰撞完成的，分子相遇后，先形成一种过渡态物种——活化络合物；b活化络合物很不稳定，一方面与反应物建立动态平衡，另一方面可分解成产物；c活化络合物分解成产物的步骤是整个反应的决速步；d 活化络合物分解的速率决定与活化络合物的浓度和性质。

优点：a 形象地描绘了基元反应进展的过程；b 原则上可以从原子结构的光谱数据和势能面计算宏观的反应速率常数；c对Arrheinus公式的指前因子作了理论说明，认为它与活化熵有关；d用势能面形象的说明了为什么需要活化能以及反应遵循的能量最低原理。

缺点：a引进了平衡假设和速决步假设并不能符合所有的实验事实；b活化络合物的结构现在还无法从实验上确定，在很大程度上具有猜测性； c 计算方法过于复杂，在实际应用上还存在很大困难，尤其是对于复杂的多原子反应；d 绘制势能面有困难，使该理论受到一定的限制。

生态方程的稳定性分析

生态方程的稳定性分析生态方程是指用数学模型描述生态系统中物种相互作用、能量流动和物质循环的方程式。

它的稳定性分析是生态学研究的核心内容之一。

在生态系统中，各个物种之间存在着各种不同的关系。

这些关系如何相互作用，将对生态系统的稳定性产生重要影响。

生态系统的稳定性是指对外界扰动的抵抗能力。

生态系统中的物种数量、种群密度、食物链等变化，都可能对系统稳定性产生重要影响。

生态方程模型的稳定性分析，主要是通过对模型中的各个变量和参数进行研究，找到系统内各个物种和物种间的相互作用关系，从而预测生态系统的稳定性。

在生态方程模型中，物种之间的相互作用关系通常分为两大类，即竞争关系和捕食关系。

竞争关系是指不同物种之间争夺同一个资源，造成了生态压力，引起种群数量的变化。

而捕食关系则是指食物链上的物种之间通过捕食行为来互相影响。

捕食关系可以促进资源的转化和循环，从而维持生态系统的稳定性。

生态方程模型的稳定性分析中，对模型参数的灵敏度分析是一项重要内容。

生态系统中存在着许多的参数，例如物种丰富度、种群分布、环境因素等，这些参数对于生态系统的运行状态和稳定性都有着重要的影响。

灵敏度分析是在变量值发生变化时，生态方程模型的稳定性如何受到影响的研究。

具体而言，就是对每个阈值进行变化，观察对应参数变化对生态系统的影响程度。

通过这种方式，可以预测生态系统对不同干扰的抵抗能力。

同时，还可以发现导致生态系统不稳定的参数，及时提出措施加以遏制。

生态方程的稳定性分析一直是生态学研究的核心内容之一。

随着生态研究的不断深入和发展，各个研究领域之间的融合，生态方程的稳定性研究也将得到不断的推进。

同时，各国政府和社会公众也越来越重视生态系统的健康，认识到稳定性分析对生态保护的重要性。

吉布斯---亥姆霍兹方程1

• 若△G=0，体系处于平衡状态。
精选版课件ppt
7பைடு நூலகம்
吉布斯—亥姆霍兹方程
• 知道一个温度下的△rGm • 求其他温度下的△rGm？ • 求助于Ｇｉｂｂｓ－Ｈｅｌｍｈ
ｏｌｚ公式
精选版课件ppt
8
精选版课件ppt
9
精选版课件ppt
1
吉布斯—亥姆霍兹方程
• 耶鲁大学（Yale University）,旧译“耶劳大书院”，是一所坐落于美国康涅狄格州纽黑文市的私立大学，创于1701 年，初名“大学学院”（Collegiate School）。耶鲁大学是美国历史上建立的第三所大学，世界大学排名，仅次于剑桥大学和哈佛大学。
通常的化学反应是在等温等压且做体积功的条件下进行的。吉布斯—亥姆霍兹公式可以推导此条件化学反应自发方向的判据。
精选版课件ppt
6
吉布斯—亥姆霍兹方程
• 若△H﹤0, △S﹥0,则△G必为负值，而焓减、熵增，都有利于过程向正方向自发。因此△G﹤0，必是向正向自发过程。
• 若△H ﹥0，△S ﹤ 0，则△G必为正值，而焓增熵减都使正向不自发，因此△G﹥0，必是正向不自发的过程，而它的逆过程（△G﹤0）自发生的
△G=△H－T△S
该式是由吉布斯（Gibbs）和亥姆霍兹（Helmholtz）各自独立证明的，故此式叫吉布斯—亥姆霍兹（Gibbs--- Helmholtz）公式。
精选版课件ppt
5
吉布斯—亥姆霍兹方程
吉布斯—亥姆霍兹方程运用
自由能判据：吉布斯—亥姆霍兹公式表明，恒温恒压下进行的化学反应的方向和限度的判据—— 自由能地变化是由两项决定：一项是焓变△H，另一项是与熵变有关的T△S。如这两个量使△G 成为负值，则正反应是一个自发反应。因此，焓和熵对化学反应进行的方向都产生影响，只是在不同条件下产生的影响的大小不同而已。

顶刊速报│马里兰大学胡良兵等高熵催化材料最新成果速览

顶刊速报│马里兰大学胡良兵等高熵催化材料最新成果速览编者按高熵合金（HEAs）是指包含5种及5种以上元素，且每种元素原子百分比大于5%，小于35%的合金。

根据熵的定义，可以将混合熵（ΔS mix）大于等于1.5R（R为摩尔气体常数）的合金称为高熵合金，ΔS mix介于1R-1.5R的合金称为中熵合金，低于1R的称为低熵合金。

高熵和中熵合金由于表现出更独特的结构，因此可以实现卓越的相稳定性和机械稳定性，其优异的性能主要源自于高熵合金不同于传统合金的特征，即所谓的四个“核心效应”，包括高熵效应、晶格畸变效应、迟滞扩散效应和“鸡尾酒”效应。

作为一个全新的概念，高熵效应可以使体系获得较大的混合熵，使合金倾向于形成固溶体，而不是金属间化合物，该特性体现了混合熵对合金相形成的贡献。

实际上，高熵合金的特点还可以扩展到金属化合物中，他们也表现出高的混合熵。

迄今为止，已经开发了高熵氧化物（HEO）、高熵硫化物（HES）、高熵硼化物（HEB）和高熵碳化物（HEC）等。

这些高熵材料具有组分和结构可调性高的优势，因此在电催化领域展现出独特的地位。

本文总结了马里兰大学胡良兵教授、东南大学沈宝龙教授及北京大学郭少军教授等课题组在高熵催化材料方面的最新进展。

1. Nano Research：碳载高熵硫化物纳米阵列实现高性能电解水电解水技术是实现可持续绿色制氢的一个关键技术，发展低成本高性能的电解水催化剂至关重要。

过渡金属硫化物由于具有良好的导电性和优异的催化性能，成为了电解水催化剂的有力候选者，但是其结构稳定性较差，在高电位下容易坍塌。

高熵金属硫化物具有高熵效应以及单相结构，可以通过多位点协同效应优化催化性能，但是为了使不同元素之间形成均相结构，传统制备高熵金属硫化物的方法均需要高温环境，因此，开发温和的方法制备高熵金属硫化物对于电解水催化剂的发展具有重要研究价值。

鉴于此，郑州大学尚会姗副研究员等人采用温和的阳离子交换策略构筑了新型碳纤维负载的Co-Zn-Cd-Cu-Mn高熵硫化物纳米阵列，得益于其多金属位点的协同效应和高熵硫化物/碳纤维的强界面键合作用，该催化剂在碱性电解质中具有优异的析氢（HER）和析氧（OER）双功能电催化性能。