Hiriart―Urruty问题与Clarke广义梯度的若干性质

亥姆霍兹方程柯西问题的求解过程
亥姆霍兹方程是一个著名的偏微分方程，描述了波动现象的传播。

柯西问题是指在给定初始条件下求解方程。

对于二维亥姆霍兹方程：
∇²u + k²u = 0
其中， u 是待求解的函数， k 是波数。

柯西问题的初始条件一般包括波函数 u 在某一时间 t=0 和空间区域内的初始值。

要解决这个问题，一般采用 Fourier 分解法。

设 u 可以分解为平面波的叠加形式：
u(x, y, t) = ∑[An cos(kn x + ln y - ωn t) + Bn sin(kn x + ln y - ωn t)]
其中， An、Bn 是待定系数， kn、ln 是波数，ωn 是与 kn 有关的频率。

将初始条件代入上述公式，可以得到 An 和 Bn 的值。

然后将其代入泛定解中，即可以得到方程的求解结果。

需要注意的是，在实际问题中，亥姆霍兹方程的求解往往还需要结合具体的边界条件来求解。

具体求解过程可能因问题的复杂性而有所不同，可针对具体问题采用适当的数值解法（如有限差分法、有限元法等）进行求解。

克拉夫特不等式克拉夫特不等式是编码理论中的一个数学关系，给出了一个码字长度集合存在唯一可解编码/单义可译码（uniquely decodable code）的必要条件。

因为这个不等式在前缀码和树上面应用很多，所以在计算机科学和信息学中很常用。

设符号表中的原始符号为：在大小为的字符集上编码为唯一可解编码的码字长度为则，反之, 给定一个满足上述不等式的自然数集合, 则在大小为字符集上，存在一组唯一可解编码符合相应的码字长度。

1949年，克拉夫特不等式发表在克拉夫特的一本专著上。

然而，克拉夫特的论文只讨论了前缀码，并将这个不等式的归因于雷蒙德·雷德赫弗不等式（Raymond Redheffer）。

1956年，麦克米兰独立发现了这个结果。

他证明了一般情况下唯一可解码的结果，并将前缀码的版本归因于Joseph Leo Doob在1955年提到的一种现象。

应用克拉夫特不等式对码字限制长度以保证前缀编码的可能性。

这个不等式说明码字长度指数的倒数的分布和概率质量函数很相似。

克拉夫特不等式可以想象成一个受限的编码库，越短的编码代价越大。

如果克拉夫特不等式中严格成立，相应的编码有冗余（redundancy）。

如果克拉夫特不等式中等式成立，相应的编码被称作完备码（complete code）。

如果克拉夫特不等式不成立，相应的编码不是唯一可解编码（uniquely decipherable）。

对于每一个唯一可解码的代码，都存在一个长度分布相同的前缀码。

影响及意义克拉夫特不等式(Kraft inequality)信源编码理论中的一个重要不等式.当一个码的任意码字与比它更长的任意码字的字首不相同时，称此码为满足字首条件的码.由码字分别为N;(i=1,2,……,M)的M个码字所组成而且又满足字首条件的码，其存在的充分必要条件是满足公式M-N ＜1此式称为克拉夫特不等式.craft不等式是克拉夫特不等式，克拉夫特不等式(Kraftinequality)信源编码理论中的一个重要不等式，当一个码的任意码字与比它更长的任意码字的字首不相同时，称此码为满足字首条件的码。

M OLECULAR AND C ELLULAR B IOLOGY,Nov.2007,p.7955–7965Vol.27,No.22 0270-7306/07/$08.00ϩ0doi:10.1128/MCB.00908-07Copyright©2007,American Society for Microbiology.All Rights Reserved.Loss of Emi1-Dependent Anaphase-Promoting Complex/Cyclosome Inhibition Deregulates E2F Target Expression and Elicits DNADamage-Induced Senescenceᰔ†Emmy W.Verschuren,1Kenneth H.Ban,2Marilyn A.Masek,2Norman L.Lehman,2and Peter K.Jackson1,2*Department of Tumor Biology and Angiogenesis,Genentech Inc.,1DNA way,South San Francisco,California94080,1and Department of Pathology,Stanford University School of Medicine,Stanford,California943052Received22May2007/Returned for modification25June2007/Accepted4September2007Expression of the anaphase-promoting complex/cyclosome(APC/C)inhibitor Emi1is required for theaccumulation of APC/C substrates crucial for DNA synthesis and mitotic entry.We show that in vivo Emi1expression correlates with the proliferative status of the cellular compartment and that cells lacking Emi1undergo cellular senescence.Emi1depletion leads to strong decreases in E2F target mRNA and APC/Csubstrate protein abundances.However,cyclin E mRNA and cyclin E protein levels and associated kinaseactivities are increased.Cells lacking Emi1undergo DNA damage,likely explained by replication stress uponderegulated cyclin E-and A-associated kinase activities.Inhibition of ATM kinase prevents induction ofsenescence,implying that senescence is a consequence of DNA damage.Surprisingly,no senescence or noextensive amount of senescence is evident upon depletion of the Emi1-stabilizing factor Evi5or Pin1,respec-tively.Our data suggest that maintenance of a protein stabilization/mRNA expression positive-feedback circuitfueled by Emi1is required for accurate cell cycle progression,maintenance of DNA integrity,and preventionof cellular senescence.The timely transcriptional activation and protein stabiliza-tion of cell cycle regulators are crucial for irreversible anderror-free cell cycle progression.During G1,these events arelimited by the retinoblastoma(Rb)family of proteins,which repress E2F-dependent transcription(11),and by the ana-phase-promoting complex/cyclosome(APC/C),which drives the ubiquitin-dependent proteolysis of cyclins(39).Proteinaccumulation at G1/S therefore ultimately requires inactivationof the Rb protein through phosphorylation by cyclin-depen-dent kinases.APC/C activity is inhibited by Emi1to permit stabilization of key substrates,including the mitotic cyclins A and B(20).Importantly,Emi1is itself an E2F target gene,thus bridging transcription and protein stabilization.Emi1protein expression persists from G1/S until early mi-tosis.Its degradation in prometaphase is triggered upon se-quential phosphorylation by cyclin B/Cdk1and Polo-like ki-nase1(Plk1)kinases,thereby generating a recognition motif for the SCF␤TrCP E3ubiquitin ligase(18,30,36).A pool of Emi1remains expressed at the spindle poles beyond prometa-phase to organize spindle pole focusing through the END(Emi1/NuMa/dynein)network(1).During G2and early mito-sis,Plk1and Cdk kinases are active,and during this time,Emi1 stability is ensured through two proposed mechanisms:binding of Evi5protein to Emi1(16)and binding of the Pin1peptidyl-prolyl cis/trans isomerase to Emi1(5).Both of these mechanisms obstruct the binding of␤TrCP to Emi1,thereby protecting Emi1from precocious degradation.The cell cycle expression pattern of Emi1protein in somatic cells already points to cellular functions for Emi1in G1/S-and M-phase progression.The biological function of Emi1has been further studied by ectopic expression of a stable form of Emi1,which results in a stabilization of APC/C substrates, prolonged prometaphase,and eventual mitotic catastrophe (30).This proliferative block seen upon Emi1overexpression is absent in cells lacking p53,allowing for a further increase in genomic instability(26).In addition,loss of Emi1was shown to result in a decrease in S-phase cells,presumably because of decreased cyclin A accumulation(20).Loss of Emi1also leads to rereplication as a consequence of decreased levels of cyclin A and geminin APC/C substrates,both inhibitors of replication origin licensing(27).Importantly,a recent Emi1-knockout ap-proach showed that embryos lacking Emi1do not survive be-yond embryonic day7.5and manifest defects in mitosis,while polyploid trophoblast giant cells were unaffected(25).To-gether,thesefindings highlight a crucial role for regulation ofAPC/C activity by the Emi1protein in both G1/S and mitotic cell cycle phases.Here we studied the pattern of Emi1expression in mouse tissues and show that Emi1is specifically expressed in prolif-erating Ki67-positive compartments of the hair follicle,sper-matogonia,and intestinal crypts.Furthermore,a strict cor-relation exists between Emi1expression levels and the proliferative status of cultured cells.In addition,we show that although depletion of Emi1leads to a general decrease inexpression of G1/S markers,including cyclin A mRNA and protein levels,this is accompanied by an unexpected increase in cyclin E message,protein,and associated kinase activities.*Corresponding author.Mailing address:Department of Tumor Biology and Angiogenesis,Genentech Inc.,1DNA way,South San Francisco,CA94080.Phone:(650)467-8655.Fax:(650)225-4000.E-mail:PJackson@.†Supplemental material for this article may be found at http://mcb/.ᰔPublished ahead of print on17September2007.7955Thisfinding places cyclin E gene transcription in a category separate from other E2F target messages,potentially implying a previously uncharacterized cellular compensatory response.We speculate that this unbalanced G1/S kinase activity un-leashes a replication stress response,and wefind that DNA damage precedes eventual cellular senescence in Emi1-de-pleted cells.Importantly,senescence can be prevented by ATM inhibition,and both DNA damage and senescence re-sponses are more prominent than rereplication upon Emi1 depletion.No such senescence is seen upon Evi5depletion, emphasizing that Emi1,but not necessarily its regulators,links APC/C regulation with DNA damage-induced senescence.To-gether,our data suggest a crucial in vivo role for Emi1in E2F target mRNA and protein accumulation,the coordination of replication with mitosis,and prevention of DNA damage-in-duced cellular senescence.MATERIALS AND METHODSCell lines and treatment.HeLa,HCT-116,and U2OS cells were from ATCC and were maintained in Dulbecco’s modified Eagle’s medium(GibcoBRL)ac-cording to standardized procedures.hTERT-RPE1cells were obtained from Clontech and maintained according to the manufacturer’s recommendations.C1 human SNF5-inducible malignant rhabdoid tumor cells were a kind gift from P. Verrijzer(Leiden,The Netherlands)and were cultured as described previously (38).Where indicated,cells were synchronized in mitosis by treatment with100 ng/ml nocodazole(Sigma)for18h,collected by shake-off,washed with phos-phate-buffered saline(PBS),and replated in nocodazole-free medium.DNA damage was induced by treatment with1␮M etoposide(Sigma).ATM kinase inhibitor(KU-55933;Calbiochem)was added to afinal concentration of10␮M. Antibodies and immunoblotting.Bacterially produced maltose-binding pro-tein–mouse Emi1fusion protein was used to raise polyclonal antibodies in rabbits(Josman,LLC).Antibodies were affinity purified against glutathione S-transferase–mouse Emi1N terminus(amino acids1to219)and verified by using immunoblotting and immunocytochemistry methods(see the supplemental material).Antibodies from two of four rabbits gave comparable and mouse Emi1-specific results.Anti-human Emi1(20)and anti-human Evi5(16)antibod-ies were described previously.For immunoblotting,cell lysates were prepared in RIPB lysis buffer(100mM NaCl,50mM␤-glycerophosphate,5mM EDTA, 0.1%Triton X-100,1mM dithiothreitol,and protease inhibitors),and5to10␮g was analyzed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis.An-tibodies used for immunoblotting analysis were p21(Pharmingen),p27Kip1 (Zymed),p45Skp2(Zymed),phospho-Rb(Ser807/811;Cell Signaling),FLAG-M2 (Sigma),and Cdh1(Lab Vision Corp.).Actin(I-19),cyclin A(H-432),cyclin E (HE12),E2F1(C-20),p16(H-156),geminin(FL-209),and p53(DO-1)antibodies were from Santa Cruz Laboratories.Flow cytometry.To assess cellular DNA content,cells were washed twice with PBS andfixed in cold70%ethanol.Cells were then resuspended in PBS con-taining50␮g/ml propidium iodide(Sigma),200␮g/ml RNase A(Calbiochem), and0.1%glucose and immediately analyzed byflow cytometry on a FACScan cytometer(Becton Dickinson)using CellQuest software.siRNA transfections.Small interfering RNA(siRNA)duplexes(Dharmacon) were transfected at100nMfinal concentrations by using Oligofectamine reagent (Invitrogen)according to the manufacturer’s instructions.Target sequences were5ЈACUUGCUGCCAGUUCUUCA3Ј(for human Emi1),5ЈGCAGAAGCCAUU AUGGGUU3Ј(for human Evi5),and5ЈGCAACGATGTGTCTCCCTATT3Ј(for human Cdh1).Controls were transfections with siRNA duplexes targeting green fluorescent protein sequence(5ЈGGCTACGTCCAGGAGCGCACC3Ј).RT-PCR analysis.Total RNA from cells transfected with siRNAs was pre-pared using the RNeasy Mini system(QIAGEN)following the provided instruc-tions.Template RNA(10ng)was analyzed using SuperScript one-step reverse transcriptase PCR(RT-PCR)analysis(Invitrogen)following the manufacturer’s recommendations.Briefly,primers were used at0.2␮M,cDNA synthesis was for 30min at50°C,and annealing temperatures were typically10°C below primer melting temperatures.RT analysis was semiquantitative in that samples were taken every two cycles and results were analyzed at below-saturation signal intensities(typically28to33cycles).Primers all spanned exon-intron boundaries to prevent signal interference resulting from DNA priming.Forward and reverse primer sequences were as follows:for Emi1,5ЈTGTTCAGAAATCAGCAGC CCAG3Јand5ЈCAGGTTGCCCGTTGTAAATAGC3Ј(200nucleotides[nt]); for Evi5,5ЈGAGATGGAAAGACCCACCCAAG3Јand5ЈTTGTCGTAGTT CAGCCACAGCAGC3Ј(350nt);for cyclin A,5ЈAGACCCTGCATTTGGCT GTGAA3Јand5ЈACAAACTCTGCTACTTCTGG3Ј(150nt);for glyceralde-hyde-3-phosphate dehydrogenase,5ЈTGGAAATCCCATCACCATCT3Јand5ЈTTCACACCCATGACGAACAT3Ј(200nt);for Plk1,5ЈCCAGAGGGAGA AGATGTCCA3Јand5ЈATAACTCGGTTTCCGTGCAG3Ј(ϳ300nt);for cyclin E,5ЈGGAGCCAGCCTTGGGACAATAATG3Јand5ЈTGTCACATA CGCAAACTGGTGCAAC3Ј(580nt);for E2F1,5ЈCATTGCCAAGAAGTC CAAGAACC3Јand5ЈATGCTACGAAGGTCCTGACACG3Ј(250nt);and for p16,5ЈCAGACATCCCCGATTGAAAGAAC3Јand5ЈCTCACTCCAGA AAACTCCAACACAG3Ј(300nt).Immunoprecipitation and kinase assays.Cell lysates were prepared in RIPB buffer,and500␮g of total protein was incubated for2h with2␮g of antibodies to cyclin E(C-19;Santa Cruz),cyclin A(Upstate),or Cdk2(M2;Santa Cruz). Specific antigen was captured by using protein G-or protein A-Sepharose beads (Sigma)and washed using RIPB and kinase buffer(50mM NaCl,20mM HEPES,pH7.2,10mM MgCl2,2mM EDTA,0.02%Triton X-100).Beads or4 units of cyclin B/Cdc2kinase as a positive control(New England Biolabs)was next incubated with250␮g/ml histone H1substrate protein(Upstate)in kinase buffer supplemented with66␮M Na-ATP and0.25mCi/ml[␥-32P]ATP (PerkinElmer).Reaction mixtures were incubated at30°C for30min and ana-lyzed by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and autora-diography.Senescence-associated␤-galactosidase(SA-␤-gal)staining.Cellular senes-cence was detected by staining for acidic(pH6.0)␤-galactosidase activity as described previously(14).Immunohistochemistry.Immunohistochemical staining was performed on 0.4-␮m paraffin-embedded tissue sections from skin(cheek),intestines,and testes of C57BL/6mice.The sections were deparaffinized and antigen was retrieved from them by use of citrate(pH6.0)buffer and microwaving.Endog-enous peroxidase and nonspecific binding were blocked using3%hydrogen peroxide and Power Block(Biogenex),respectively.The chromogen was3,3-diaminobenzidine(Biogenex),and Mayer’s hematoxylin was used as a counter-stain.Primary antibodies were affinity-purified rabbit anti-mouse Emi1and anti-Ki67antibody(Abcam),and the secondary antibody was Envision Plus(Dako) anti-rabbit antibody–horseradish peroxidase.Immunofluorescence.Cells growing on coverslips werefixed in4%paraform-aldehyde,permeabilized with0.2%Triton X-100,and immunostained using0.4 to2␮g/ml primary antibody and Alexa Fluor488or Cy3secondary antibodies (Molecular Probes).Primary antibodies used were against PCNA(PC10;Santa Cruz),phospho-histone H2A.X Ser139(Upstate),phospho-Chk2Thr68(Cell Signaling),phospho-Ser15p53(Cell Signaling),or replication protein A(RPA; NeoMarkers).DNA was counterstained with Hoechst33288dye and image analysis was performed using a Zeiss AX10microscope and Slidebook4.1soft-ware.Microarray analysis.Cells treated with siRNA as described above were har-vested at22or76h after transfection and total RNA was prepared using the RNeasy Mini system(QIAGEN)incorporating an additional on-column DNase digestion step.The methods for preparation of cRNA and array hybridization were provided by Affymetrix(Santa Clara,CA).cRNA was hybridized to Af-fymetrix Human Genome U133Plus oligonucleotide arrays,which were scanned on a GeneChip3000scanner.Data analysis was performed using Affymetrix GCOS1.4software.Experiments were performed in triplicate,and averages were compared to those of controls by using routines in the R programming language(script available on request).RESULTSEmi1is expressed in proliferating cell compartments in vivo.Somatic cells express the Emi1protein from G1/S throughearly mitosis,thereby defining a window of stabilization and accumulation of critical S-phase and M-phase regulators(20). Together with the fact that Emi1is an E2F target gene,these data suggested that Emi1protein is expressed in proliferating cells.We therefore investigated the expression of Emi1protein in mouse tissues.Affinity-purified anti-mouse Emi1polyclonal antibody was verified to specifically recognize endogenous mouse Emi1protein(see Fig.S1in the supplemental mate-rial).Consistent with the biochemical role of Emi1in G1/S7956VERSCHUREN ET AL.M OL.C ELL.B IOL.progression,immunohistochemistry of mouse tissues con-firmed the expression of Emi1in proliferating Ki67-positive cellular compartments in the mouse hair follicle and skin epi-dermis,spermatogonia,and intestinal crypts (Fig.1A).Conversely,we asked whether Emi1expression is reduced in nonproliferating cells,either upon induction of quiescence or irreversible cell cycle exit (cellular senescence).We first stud-ied quiescent human diploid telomerase RT-immortalized ret-inal pigment epithelial (RPE)cells that were induced to re-sume cycling by serum addition.As expected,serum-starved RPE cells expressed high levels of the p27Kip1Cdk inhibitor and low levels of G 1/S cyclins,including cyclin E and cyclin A (Fig.1B).Emi1protein was undetectable in quiescent RPE cells but present at times coinciding with cyclin A protein accumulation,consistent with the notion that Emi1is required to stabilize cyclin A through APC/C inhibition.Of note,levels of the Emi1-stabilizing protein Evi5were only slightly de-creased upon quiescence.This shows that,in comparison with Emi1,Evi5expression was less significantly affected by prolif-erative status in this system (Fig.1B).The effect of expression of Emi1in cells induced to undergo irreversible cell cycle exit (senescence)was studied in two different systems.The first used IPTG (isopropyl-␤-D -thio-galactopyranoside)-inducible expression of the SNF5chromatin remodeling factor in malignant rhabdoid tumor cells (C1).These cells lack endogenous SNF5and represent a well-stud-ied example of inducible senescence that is thought to involve transcriptional induction of p16Ink4a (38).Secondly,we used RPE cells induced to undergo a DNA damage checkpoint response by etoposide treatment.In both cases,a reduction of Emi1protein levels correlated with cell cycle exit (Fig.1C and D;cell cycle exit upon etoposide treatment is shown in Fig.S2in the supplemental material),and again,Evi5protein levels were not significantly affected,on occasion showing even a slight elevation upon DNA damage.The decrease in Emi1protein levels of dimethyl sulfoxide-treated cells reflectstheFIG.1.Emi1expression in vivo correlates with cell proliferation status.(A)Immunoperoxidase stainings of mouse skin,testis,and intestine sections with anti-mouse Emi1and Ki67antibodies showing Emi1expression in proliferative compartments.Arrows indicate hair follicle outer root sheath or skin epidermis (upper panels),spermatogonia (middle panels),or intestinal crypt cells (lower panels).Inserts show magnified sections of the images.(B)RPE cells were serum starved for 7days,and cell cycle reentry was monitored by harvesting cell lysates at the indicated time points following growth in 10%serum.Cell lysates were analyzed by immunoblotting using the indicated antibodies.The circled P indicates phosphorylation.(C)Left panel,C1cells were grown for 4days in the presence or absence of 2mM IPTG to induce FLAG-SNF5expression,and cell lysates were immunoblotted with the indicated antibodies.Right panel,cells were grown for 7days in the presence of IPTG and stained for SA-␤-gal,confirming induction of cellular senescence upon SNF5expression.(D)RPE cells were grown in the presence of 1mM etoposide or solvent control (dimethyl sulfoxide [DMSO])for the indicated number of days.Cell lysates were immunoblotted with the indicated antibodies.The asterisk indicates a nonspecific cross-reacting band.V OL .27,2007Emi1LINKS APC/C INHIBITION WITH CELLULAR SENESCENCE 7957fact that proliferating RPE cells eventually become density-arrested when reaching confluence.Together,these data rep-resent thefirst analysis of Emi1expression in vivo,demonstrat-ing the presence of Emi1protein in proliferating cell compartments,and show that Emi1expression is positively correlated with proliferative status by using in vitro cell sys-tems.Emi1knockdown results in a cellular proliferation arrest and senescence.We previously showed that cells lacking Emi1 protein show a reduced S-phase fraction as measured by bro-modeoxyuridine(BrdU)incorporation and a delayed accumu-lation of cyclin A in synchronized cells(20).Here,we extended this analysis and detected a significant reduction in the prolif-erative capacity of RPE cells treated with human Emi1siRNA (Fig.2A).Analysis of the cell cycle profile by propidium iodide (PI)staining andfluorescence-activated cell sorter analysis demonstrated that cells treated with Emi1siRNA indeed un-dergo a delayed progression through S and G2phases com-pared to control siRNA transfected cells(Fig.2B;quantita-tions are shown in Fig.S3A in the supplemental material). Importantly,this delay was partially rescued by codepletion of Cdh1protein,suggesting that this is due to the destruction of APC/C substrates upon Emi1depletion(Fig.2B;see also Fig. S3A in the supplemental material).In addition,a modest increasein cells with a larger than G2DNA content was visible(up toϳ6.3%),indicating that a percentage of cells undergoes en-doreduplication.Similar results were obtained using an addi-tional Emi1siRNA targeting sequence(see Fig.S3B in the supplemental material).Of note,more significant endoredu-plication was measured upon Emi1depletion in transformed HeLa and U2OS cells(see Fig.S3C in the supplemental ma-terial),highlighting the fact that checkpoint signaling in pri-mary cells and that in transformed cells are likely different. To better assess DNA replication dynamics,we next mea-sured BrdU incorporation rates by usingflow cytometry. Consistent with the PI profiles,an increased percentage of cells were BrdU positive early after Emi1depletion,imply-ing slower S-phase progression.Emi1-depleted cells,how-ever,exhibited a reduction in the rate of S-phase entry, evidenced by reduced BrdU incorporation in synchronized cells(see Fig.S3D in the supplemental material).The latter explains published data describing reduced BrdU incorpo-ration in asynchronously growing Emi1-depleted cells(20, 27),as these studies applied longer(2-h)BrdU-labeling pulses(as opposed to a30-min pulse in our present exper-iment),allowing the detection of cells both entering and progressing through S phase.Cells lacking Emi1were large andflattened and contained relatively large nuclei(Fig.2A),a cellular morphology remi-niscent of cellular senescence.We therefore examined human Emi1siRNA-treated RPE cells for the characteristic SA-␤-gal marker.Approximately37to65%of RPE cells stained positive for SA-␤-gal at4or5days after Emi1siRNA treatment(Fig. 2C),a phenotype that was recapitulated using two additional siRNA targeting sequences(see Fig.S4A in the supplemental material).Importantly,senescence induction upon Emi1de-pletion required APC/C activity,as it was rescued upon codepletion of Cdh1(Fig.2D),and was also observed in HCT-116and HeLa cells(see Fig.S4B in the supplemental mate-rial).Quantitation of DNA content by using microscopy showed that enlarged senescent nuclei contained a relative integrated DNA intensity similar to that of control cells(see Fig.S4C in the supplemental material),confirming theflow cytometry data shown in Fig.2B.Together,Emi1depletion and consequent APC/C activation lead to the induction of senescence both in RPE cells that resemble primary cells and in transformed cells that lack critical tumor suppressor path-ways(the p53pathway in HeLa cells)or contain oncogenic mutations(the Ras pathway in HCT-116cells).Depletion of Emi1results in suppression of E2F target genes but an increase in cyclin E.To further explore the initial biochemical response to Emi1depletion,we followed protein expression in synchronized control or Emi1siRNA-treated cells.As shown previously,Emi1knockdown results in a de-creased accumulation of cyclin A when cells are followed from mitosis to G1/S(Fig.3A).The decreased cyclin A accumula-tion correlated with a decreased phosphorylation and there-fore a transcriptionally repressive form of the Rb pocket pro-tein.Correspondingly,decreased abundance of the E2F target protein and APC/C substrate Skp2was measured(4,49),as well as a decrease in E2F1protein,itself an E2F target gene product.In addition,we measured a decrease in the APC/C target protein geminin,an inhibitor of replication licensing (Fig.3A).Emi1-depleted cells,however,showed a marked increase in cyclin E1protein levels,an effect that was also seen by using an additional Emi siRNA oligonucleotide and seen up to48h or more after release from nocodazole(see Fig.S4D in the supplemental material).Of note,Cdh1codepletion re-stored Rb phosphorylation and cyclin A and E protein levels to control levels(see Fig.S4E in the supplemental material), confirming that these biochemical responses to Emi1depletion are explained by an increase in APC/C activity.No significant induction of the Cdk inhibitor p16Ink4A was seen,while p27Kip1 levels were only slightly increased in this time course.Taken together,Emi1knockdown results in APC/C activation and a consequent decrease in E2F target protein expression with the exception of an increase in cyclin E1protein levels.Cyclin E-directed Cdk2and GSK3kinase activities together mediate cyclin E phosphorylation and subsequent ubiquitina-tion in an autoregulated destruction loop(10,50,52).It was therefore surprising tofind high levels of cyclin E protein without concomitant increases in its inhibitor p27Kip1.To bet-ter understand this result,we analyzed the abundance of cell cycle mRNAs by RT-PCR.Again,an overall reduction in E2F target gene transcription,including transcription of cyclin A, E2F1,and Plk1genes,was measured,while cyclin E gene transcript levels were significantly increased(Fig.3B).This result was confirmed by microarray analysis of RNAs isolated at22or76h after Emi1siRNA treatment(Fig.3C),which also showed decreases in cyclin B and D1mRNA abundances.A similar increase in cyclin E mRNA levels was seen with the use of an additional Emi1siRNA oligonucleotide,emphasizing that this is an on-target effect(see Fig.S4G in the supplemen-tal material).Of note,while p16Ink4A induction is often mea-sured upon cellular senescence,we see no increase in p16Ink4A mRNA at22h and only a slight induction at72h.Thus,Emi1 expression and consequent APC/C inhibition are required for the accumulation of E2F target proteins and transcripts,with cyclin E as a notable exception.We next asked whether this increase in cyclin E level corre-7958VERSCHUREN ET AL.M OL.C ELL.B IOL.lates with an increase in cyclin E/Cdk2kinase activity.Cell extracts were subjected to cyclin E or Cdk2immunoprecipita-tions,and associated kinase activities were assessed in vitro.Cell extracts from Emi1-depleted cells showed a significant increase in both cyclin E-and Cdk2-associated kinase activi-ties,whereas cyclin A-associated activity was reduced (Fig.3D).Together,these data imply that G 1/S progression and cyclin/Cdk activities are deregulated in cells lackingEmi1.FIG.2.Cells lacking Emi1undergo a proliferation arrest and cellular senescence.(A)Left panel,growth curves of control or Emi1siRNA-treated RPE cells.Cells were plated at equal cell numbers and transfected with the indicated siRNAs at day 0.Cells were counted at days 0to 5after transfection.Averages and standard deviations of three individual experiments are shown.Right panel,RPE cells were transfected with control or Emi1siRNA and microscopically analyzed 2or 5days later.Fewer and larger cells were seen after Emi1knockdown.(B)Cells treated with control,Emi1,or Emi1and Cdh1siRNAs were harvested on the indicated days,stained with PI,and analyzed by flow cytometry to establish cell cycle profiles.FL2-A,PI fluorescence.(C)RPE cells transfected with control or Emi1siRNA were grown for 4or 5days and then stained for SA-␤-gal marker to establish the status of cellular senescence.The percentages of blue,senescent cells are indicated (200to 300cells in total).(D)RPE cells were transfected with Emi1or Cdh1siRNA alone or a combination of Emi1and Cdh1siRNAs.Cells were grown for 5days and then stained for SA-␤-gal.The percentages of blue,senescent cells are indicated (200cells in total).The lower panel shows Western analysis of lysates harvested 2days after transfection and confirms knockdown of Emi and Cdh1proteins.V OL .27,2007Emi1LINKS APC/C INHIBITION WITH CELLULAR SENESCENCE 7959Replication stress and DNA damage induction upon Emi1knockdown.Recent work has indicated that deregulated or increased cyclin E/Cdk kinase activities are associated with altered replication dynamics and consequent DNA damage induction (2).Stalled replication leads to the formation of single-stranded DNA intermediates that are visualized by re-cruitment of RPA (56).We measured a significant increase in cells with distinct nuclear RPA foci upon Emi1depletion (Fig.4A),suggesting replication stress.Premature termination of DNA replication can lead to fork collapse and consequent DNA double-strand breaks,eventually triggering robust acti-vation of the DNA damage checkpoint (43).Indeed,staining of Emi1-depleted cells showed nuclear foci containing phos-phorylated histone H2AX (␥-H2AX),a marker of DNA dam-age foci (Fig.4B).In addition,a significantly increased per-centage of cells contained the active phosphorylated form of the Chk2checkpoint kinase (Fig.4C),implying activation of the upstream ATM/ATR kinases (31,43).The amount and size of foci stained with either ␥-H2AX or phospho-Chk2per individual nucleus were measurably greater in Emi1siRNA-treated cells than those in control siRNA-treated cells,and these two DNA damage markers colocalized in all nuclei that contained the highest numbers of foci (Fig.4E).Thus,Emi1depletion elicits a potent DNA damage response.Active ATM/ATR and Chk kinases initiate a phosphory-lation cascade aimed at halting cells and executing DNA repair processes.This includes serine 15phosphorylation of the p53protein,resulting in its stabilization and transcrip-tional activation (41).Indeed,Emi1siRNA-treated cells contained an increased percentage of cells with nuclear phospho-Ser15-p53(Fig.4D),which also correlated with the extent of ␥-H2AX focus formation (Fig.4F).Activation of p53was further evidenced by an increased expression of the p53target protein p21as early as 2days after siRNA treat-ment (Fig.4G).The data described above suggest that loss of Emi1results in DNA replication stress,likelyinvolvingFIG.3.Reduced E2F target expression but increased cyclin E levels in Emi1-depleted cells.(A)RPE cells were transfected with control or Emi1siRNA and 4h later treated with nocodazole (noc.)for 18h to arrest cells in mitosis.Mitotic cells were collected by shake-off and grown in nocodazole-free medium for the indicated numbers of hours.Proteins were analyzed by immunoblotting with the indicated antibodies.The circled P indicates phosphorylation.(B)Cells treated as described for panel A were harvested at 18h after release from mitosis.Total RNA was prepared and analyzed by RT-PCR analysis for the indicated genes.GAPDH,glyceraldehyde-3-phosphate dehydrogenase.(C)RPE cells were transfected with control or Emi1siRNA and harvested 22or 76h later.Total RNA was prepared and DNA was removed by DNase treatment.Samples were analyzed by microarray on Affymetrix HGU133P oligonucleotide chips.The difference in the average log 2[intensity]between treatment and control (corresponding to change [n -fold]due to treatment)is shown for each of two time points per gene.Asterisks indicate significant (P Ͻ0.05)differential expression between treatment and control groups.(D)Top panel,RPE cells were transfected with control (M)or Emi1siRNA (E)and harvested 22h later,and cell extracts were prepared.Proteins were immunoprecipitated by using the indicated antibodies,and associated kinase activities towards histone H1(HH1)substrate were determined using autoradiography for [␥-32P]ATP.Bottom panel,negative controls using immunoglobulin (IgG)control immunoprecipitations (IPs)and omission of H1substrate confirm signal specificity.CycA and CycE,cyclin A and cyclin E.7960VERSCHUREN ET AL.M OL .C ELL .B IOL .。

凸优化广义不等式-概述说明以及解释1.引言引言部分是文章的开篇，用于引起读者的兴趣，并简要介绍文章要讨论的主题。

在本文中，引言的概述部分应该包含凸优化和广义不等式的基本概念。

下面是一个可能的写作范例：1.1 概述在现代数学和应用领域中，凸优化和广义不等式是两个重要的研究领域。

凸优化是数学中一种重要的优化方法，其旨在寻找一个函数的全局最小值，但只需要求函数定义域上的局部极小值；而广义不等式则涵盖了多种不等式类型，如线性不等式、二次不等式、半正定规划等。

凸优化作为一种强大的数学工具，广泛应用于各个领域，如金融、运筹学、信号处理等。

它不仅可以求解实际问题，并且具有良好的理论基础和算法支持。

而广义不等式则为凸优化提供了一种重要的约束条件，通过引入不等式约束，可以更加灵活地建模和求解实际问题。

本文将详细介绍凸优化和广义不等式的基本概念，并探讨它们在实际问题中的应用。

首先，我们将介绍凸优化的基本概念，包括凸集、凸函数和凸优化问题的定义。

然后，我们会详细讨论广义不等式约束的相关知识，包括线性不等式约束、二次不等式约束和半正定规划等。

通过对凸优化和广义不等式的深入理解，我们可以更加灵活地应用它们解决实际问题，提高问题求解的效率和精度。

同时，我们也能够更好地理解优化问题背后的数学原理和算法思想。

在现代科学技术的发展中，凸优化和广义不等式将发挥着越来越重要的作用，在实践中具有广阔的应用前景。

接下来，我们将开始介绍凸优化的基本概念，为后续的讨论做好准备。

1.2 文章结构文章结构的目的是为了让读者更好地理解和掌握凸优化广义不等式的相关知识。

为了达到这个目的，本文分为引言、正文和结论三个部分。

引言部分首先对该主题进行了概述，介绍凸优化和广义不等式的基本概念及其在实际问题中的应用。

接着，引言部分明确了本文的结构和目的，让读者知道本文的内容和安排。

正文部分是本文的重点，它分为两个小节：凸优化的基本概念和广义不等式约束。

在凸优化的基本概念部分，将介绍凸优化的定义、性质和常见算法。

IOP P UBLISHING N ANOTECHNOLOGY Nanotechnology20(2009)195601(9pp)doi:10.1088/0957-4484/20/19/195601A thermal study on the structural changes of bimetallic ZrO2-modified TiO2 nanotubes synthesized using supercritical CO2R A Lucky and P A Charpentier1Department of Chemical and Biochemical Engineering,University of Western Ontario,London,ON,N6A5B9,CanadaE-mail:pcharpentier@eng.uwo.caReceived4November2008,infinal form19February2009Published21April2009Online at /Nano/20/195601AbstractIn this study the thermal behavior of bimetallic ZrO2–TiO2(10/90mol/mol)nanotubes arediscussed which were synthesized via a sol–gel process in supercritical carbon dioxide(scCO2).The effects of calcination temperature on the morphology,phase structure,mean crystallite size,specific surface area and pore volume of the nanotubes were investigated by using a variety ofphysiochemical techniques.We report that SEM and TEM images showed that the nanotubularstructure was preserved at up to800◦C calcination temperature.When exposed to highertemperatures(900–1000◦C)the ZrO2–TiO2tubes deformed and the crystallites fused together,forming larger crystallites,and a bimetallic ZrTiO4species was detected.These results werefurther examined using TGA,FTIR,XRD and HRTEM analysis.The BET textural propertiesdemonstrated that the presence of a small amount of Zr in the TiO2matrix inhibited the graingrowth,stabilized the anatase phase and increased the thermal stability.(Somefigures in this article are in colour only in the electronic version)1.IntroductionConsiderable effort is being devoted to the preparation of one-dimensional(1D)oxide nanostructures due to their poten-tial applications in a diversity of technologies including catal-ysis,high efficiency solar cells,coatings and sensors[1–3].In particular,titania(TiO2)nanotubes are receiving considerable attention due to titania’s unique optoelectronic,photochemical and dielectric properties,along with being a low cost material for potential commercial employment[4–7].Having such unique properties,TiO2nanomaterials with defined structures are highly desirable for electron-transport materials in dye-sensitized solar cells,as photocatalysts,photoconductive agents and nanofillers in polymer composites[7,8].The performance of these titania nanomaterials widely relies on their crystallinity,crystallite size,crystal structure,specific surface area and thermal stability[9,10].As examples,the 1Author to whom any correspondence should be addressed.role of anatase and rutile crystal phases in TiO2nanostructures is still under active investigation for photocatalysis,where the interface between these nanostructures is believed to be the catalytic active site[11].The rutile crystal structure is important for stronger materials for use in orthopedic applications such as bone cements[12].For preparing these oxide nanostructures,the sol–gel method is becoming the standard method as it provides a uniform phase distribution,high purity,low temperature processing,and better size and morphology control[13]. However,the properties of sol–gel-synthesized binary metal oxides strongly depend on the synthesis conditions,such as the type of alkoxide(s),temperature,catalyst,solvent and solvent removal process[14,15].In the past decade,direct sol–gel reactions in supercritical carbon dioxide(scCO2)have attracted much attention for synthesizing oxide nanomaterials[16–19]. This approach has many advantages over the conventional sol–gel process operated in an organic solvent,as the resulting ma-terials maintain nanofeatures and a high surface area after CO2drying and venting[20].Low viscosity,‘zero’surface tension and high diffusivity of scCO2are favorable physical properties of the solvent for synthesizing potential superior ultrafine and uniform nanomaterials.As well,CO2is inexpensive, inflammable and considered a‘green’reaction medium[21].Previously,we discovered that bimetallic Zr-modified TiO2(Zr–TiO2)nanotubes could be synthesized in scCO2,and our preliminary results indicated that the nanotubes gave a high surface area,up to430m2g−1,as prepared[22].According to several studies,a small amount of transition metal doping has been found very effective to improve the thermal stability and activity of TiO2,particularly by using zirconia[23–26].In addition to the synthesis conditions,the calcination conditions are very important to the crystal structures of the metal oxide nanomaterials obtained,and subsequently,their potential end use applications.Spijksma et al[27]synthesized titania–zirconia microporous composite membranes using a1:1molar ratio by using the sol–gel process.The crystallization temperature for these materials was750◦C;however,after calcining at800◦C,an orthorhombic ZrTi2O6structure was formed,commonly known as srilankite.Whereas Zou et al [28]synthesized binary oxides by hydrolysis of titanium and zirconium nitrate solutions at various ratios.After calcining at 800◦C,the binary oxide showed the presence of the ZrTiO4 crystal phase and very low surface areas.Kitiyanan et al [23]synthesized5mol%zinconia-modified TiO2using a sol–gel process.They showed that this small amount of Zr stabilized the anatase phase up to800◦C,but that the anatase phase completely transformed into the rutile phase at 1000◦C.However,the materials calcined at these very high temperatures showed very low surface areas.As the nanotubular structure of the bimetallic ZrO2/TiO2 nanotubes has many potentially interesting applications, however,their structure and crystal morphology changes with calcination temperature have not been investigated.Hence, this study focused on the thermal behavior of the synthesized Zr–TiO2(10/90mol/mol)nanotubes prepared via an acetic acid modified sol–gel process in scCO2.The synthesized materials were calcined at different temperatures and the effects of calcination temperature on the morphology,phase structure,mean crystallite size,specific surface area and pore volume were investigated using a variety of physicochemical characterization techniques.2.Experimental details2.1.MaterialsReagent grade titanium(IV)isopropoxide(TIP,97%,Aldrich), zirconium(IV)propoxide(ZPO,70%,Aldrich),acetic acid(99.7%,Aldrich)and instrument grade carbon dioxide (99.99%,BOC)were used without further purification.2.2.Nanotube synthesisThe procedure previously reported[22]was used for synthesis of10%ZrO2–90%TiO2nanotubes.2.3.CharacterizationScanning electron microscopy(SEM)measurements were usedto determine the size and morphology of nanomaterials usinga LEO1530scanning electron microscope.Transmissionelectron microscopy(TEM)and HRTEM images wereobtained using a Philips CM10and JEOL2010f.Thespecimens were dispersed in methanol and placed on a coppergrid covered with a holey carbonfilm.Thermo-gravimetricanalysis(TGA)was performed under nitrogen atmosphere ona TA Instruments TA-Q500at a heating rate of10◦C min−1 from room temperature to1000◦C.IR spectra were recordedon a Bruker IFS55Spectrometer in the range500–4000cm−1.Each spectrum was recorded at4cm−1resolution using500scans.Sample pellets were obtained from the powder calcinedat various temperatures by mixing with a small amount of KBr,then analyzing in transmission mode.Bulk composition wasdetermined using energy-dispersive x-ray spectroscopy(EDX)attached to a LEO1530scanning electron microscope(SEM).X-ray diffraction(XRD)was performed utilizing a Rigakuemploying Cu Kα1+Kα2=1.54184˚A radiation with a power of40kV–35mA for the crystalline analysis.The broad-scan analysis was typically conducted within the2θrangeof10◦–80◦.The samples were further analyzed using a RenishawModel2000Raman spectrometer equipped with a633nmlaser.The power at the sample varied between0.2and0.5mWwith the beam defocused to an area of approximately5–10μmin diameter.The textural characterization,such as surface area,pore volume and pore size distribution of the aerogels andthe oxides,was obtained by N2physisorption at77K usinga Micromeritics ASAP2010.Prior to the N2physisorption,the samples were degassed at200◦C under vacuum.Fromthe N2adsorption isotherms,the specific surface area wascalculated.The mesopore volume(V BJH),the average pore diameters(d p)and the pore size distributions were estimated by the Barret–Joyner–Halenda(BJH)method applied to the desorption branch of the isotherm.3.Results and discussion3.1.Electron microscopy(SEM/TEM)The effects of calcination temperature on the morphology of the Zr–TiO2nanotubes’shape and size were characterized by SEM and TEM analysis.In the SEM analysis for the as-prepared materials,it can be seen infigure1(a)that the aerogel powders were composed of a one-dimensional structure,with the nanotubes having a diameter of50–140nm and a length of several micrometers.Throughout the course of heat treatment, phase changes(amorphous to anatase to rutile)and sintering phenomena of the nanotubes were revealed by SEM and TEM investigations.The SEM image infigure1(b)shows that the material calcined at500◦C has a similar structure,although very small holes are visible in the TEM image(figure2(b))on the walls of the nanotubes.The morphology of the calcined nanotubes at800◦C is still preserved,as shown infigure1(c). As the temperature was increased further to1000◦C,the initial nanotubes disappeared and were replaced by nanometer-sizedFigure1.SEM:Zr–TiO2nanotubes calcined at:(a)as-prepared,(b)500,(c)800and(d)1000◦C(bar represents200nm;all the samples were examined after platinum coating).aggregated particles in the50–100nm size range,as shown in figure1(d).Along with SEM,TEM images gave more detailed morphological information on the tubular structure.The TEM image infigure2(a)indicates that the as-prepared nanotubes possessed uniform inner and outer diameters,having thicknesses approx.14–50nm along their length,depending on synthesis conditions.Upon heat treatment to500◦C, the chemically bonded organic layer was removed from the synthesized nanotubes,resulting in small holes on the tube wall,which was confirmed by the TEM image in figure2(b),although the internal structure was still maintained at this temperature.The SEM images showed that the outer morphology was preserved at800◦C,although at900◦C the TEM images shown that the inner hole has almost vanished with this additional heat energy(figure2(c)).The TEM images along with SEM images reveal that the nanotubes were deformed and the crystallites were fused together when the calcination temperature was increased to1000◦C (figure2(d)).3.2.Decomposition behavior(TGA/FTIR)Thermo-gravimetric analysis(TG–DTG)analysis was carried out to study the thermal decomposition behavior of the synthesized Zr–TiO2nanotubes.Figure3shows three main peaks in the TG–DTG analysis,which are in the ranges of 20–120,120–250and250–500◦C.Thefirst stage with peak maxima at37◦C gave only6wt%loss,which we attribute to the removal of residual solvent present in the synthesized materials.The second peak with its maximum at200◦C is attributed to the removal of bounded water and chemically bonded organic material,with approx.19wt%lost at this stage. The third peak,with its maximum at341◦C,is broad and is attributed to the removal of any bonded/coordinated organic material and−OH groups,with approximately an additional 21wt%loss at this temperature.The weight loss over500◦C was extremely small(0.14%)and attributed to removal of bounded−OH groups.The total weight loss measured from the TG curve was46wt%.Elemental analysis(EDX)was also performed(see table1)to investigate the change of composition with calci-nation temperature.It showed that the as-prepared nanotubes contained approx.30%carbon,with this value decreasing upon increasing calcination temperature.At300◦C,the carbon content was about18%,while when the temperature was increased to500◦C,all carbon-containing organic material was removed,consistent with the TG–DTG results.Materials calcined at higher temperatures had only metal,oxygen and a ratio of oxygen to metal atom change with temperature. Surfaces of metal oxides consist of unsaturated metal and oxide ions,and are usually terminated by−OH groups.Figure 2.TEM:Zr–TiO 2nanotubes calcined at (a)as-prepared,(b)500,(c)900and (d)1000◦C.Figure 3.Weight loss of nanotubes as a function of temperature.The −OH groups are formed by dissociative adsorption of H 2O molecules to reduce the coordinative unsaturation of the surface sites.It is very difficult to analyze the amount of oxygen bonded with metal atoms only by the EDX method,as the amount of H present in the materials cannot be determined by EDX due to the low atomic weight of H.Infrared spectroscopy is an excellent method to study the behavior and properties of metal oxides [28].The powder ATR-FTIR spectra of the Zr–TiO 2nanotubes calcined at different temperatures in air are given in figure 4.TheTable positional change with calcination temperature determined by EDX.Sample(Cal.temp ◦C)Carbon (at.%)Oxygen (at.%)Ti (at.%)Zr (at.%)As-prepared 30.6±258.5±19.9±0.51.3±0.2T-30018.5±362.7±116.5±12.3±0.5T-5000±379.6±317.9±0.52.5±0.5spectra (figure 4(a))for the as-prepared nanotubes shows a broad peak at 3400cm −1assigned to the −OH group of absorbed water [29].The peaks at 1548and 1452cm −1are due to symmetric and asymmetric stretching of the zirconium titanium acetate complex,respectively [30].This metal acetate complex confirms that the acetic acid formed bridging complexes with the metal ions,helping to stabilize the structures during their synthesis and self-assembly into nanotubular structures in scCO 2.The −CH 3group contributes the small peak at 1343cm −1,while the two small peaks at 1037and 1024cm −1correspond to the ending and bridging −OPr groups,respectively [31],indicating that unhydrolyzed −OPr groups were present in the as-prepared materials [32].The oxo bonds can be observed by the bands present below 657cm −1[30].Calcination at 400◦C significantly diminishes the intensity of the C–H stretching at 2800–3000cm −1and the zirconium titanium acetate complex band at 1548andFigure4.The powder ATR-FTIR spectra of Zr–TiO2nanotubes calcined at different temperatures.1452cm−1,spectra given infigure4(b).This indicates thatthe calcination at400◦C removes any organic material presentin the as-prepared nanotubes.No trace of IR bands fromthe organic groups was detected upon further heat treatment(figure4(c))and the broad peak at3400cm−1significantlydecreased.The nanotubes calcined at1000◦C showed onlya small band at3400cm−1(figure4(d)),indicating only asmall amount of−OH groups were still present at this hightemperature.These results are consistent with the TG–DTGmeasurements and the electron microscopy results.3.3.XRD and HRTEMIn order to examine the phase structure and crystallite size,XRD and HRTEM were used to investigate the effects of thecalcination temperature on the crystal size and phase structure.During heat treatment,the as-prepared materials transferredfrom the amorphous to anatase to rutile phases.The XRDpatterns(figure5)of all the calcined samples indicate that theZr–TiO2nanotubes consist of anatase crystal,with no rutile phase being present up to700◦C.The as-prepared materials were amorphous,while whenincreasing the calcination temperature up to400◦C,thematerial reorganized itself and the anatase particles beganto grow,resulting in crystalline material.There wasno distinct Zr peak,indicating no phase separation,andthat the Zr was integrated within the anatase crystalstructure for this composition.Previous experimentsshowed that increasing concentrations of Zr alkoxidewere incorporated homogeneously into the nanotubularstructure[22].As well,we previously prepared a crystalof Zr2Ti4(μ3–O)4(OPr)4(μ–OPr)2(μ–OAc)10using lower concentrations of acetic acid[33],showing that the Zr is partof the crystal hexamer structure.By increasing the calcinationtemperature from400to800◦C,the peak intensities increasedas well as the width of the peaks becoming narrower,indicatingan improvement of the anatase phase and simultaneously thegrowth of anatase crystallites.The XRD patterns for theobserved nanotubes indicate that no rutile phase appeared upto calcination temperatures of700◦C.It also shows thatheat Figure5.The powder XRD spectra of Zr–TiO2nanotubes calcined at different temperatures.(A–anatase,R–rutile).Table2.Crystal size and crystal structure at different calcination temperatures.Sample(Cal.temp◦C)Crystallite size(nm)Crystal structure As-prepared—AmorphousT-4009.8AnataseT-50012.5AnataseT-60015.9AnataseT-70019.5AnataseT-80021.8Anatase(88%)48.1Rutile(12%)T-90027.8Anatase(42%)69.5Rutile(58%)T-100028.3Anatase(6%)90.8Rutile(94%) treatment at800◦C forms a very small peak of the rutile phase, and further heat treatment increased the amount of rutile phase, and a new peak appeared at2θ=30.4◦,which is assigned to zirconium titanium oxide(ZrTiO4)[28].The crystallite sizes of the calcined samples are summarized in table2and were estimated from these XRD patterns using Scherrer’s equation(equation(1)):D=0.9λβcosθ(1) where D is the average nanocrystallite size(nm),λis the x-ray wavelength(1.541˚A),βis the full width at half-maximum intensity(in radians)andθis half of the diffraction peak angle.For the nanotubes calcined at400◦C,crystallite sizes of approx.9.8nm were calculated,while further heat treatment increased the crystallite size moderately.Nanotubes calcined at800◦C gave crystallite sizes up to21.8nm,resulting in smaller crystallite materials,indicating that a small amount of zirconia inhibited grain growth during heat treatment[26]. The rutile crystallite size was calculated by Scherrer’s equation using rutile(110).The obtained crystallite size was>90nm at1000◦C calcination temperature,smaller than the value reported in the literature for rutile crystallites[29],likely due to the constrained geometry of the nanotubular structure.The TEM images of the1000◦C calcined nanotubes previously showed that the crystallites were fused together,formingFigure6.HRTEM:Zr–TiO2nanotubes calcined at(a)as-prepared,(b)500and(c)1000◦C.Bar represents10nm. larger crystallites.It is known that rutile and anataseshare two and four polyhedra edges,respectively,althoughboth are tetragonal[34].Due to changes in the crystalstructure,the nanotubes’morphology deformed at highercalcination temperatures.The phase compositions of the calcined samples are alsoreported in table2and were calculated using the integratedintensities of anatase(101)and rutile(110)peaks by theequation developed by Spurr and Myers[35]:X rutile=11+K(I a/I r)(2)where I a and I r are the integrated peak intensities of the anatase and rutile phases,respectively,and the empirical constant K was taken as0.79according to Spurr and Myers.From table2 we see that the material calcined at800◦C contained only12% rutile.Further heat treatment caused a dramatic increase in both the composition and size of the rutile particles,where at 900◦C,58%of the material was converted into rutile,which increased to approx.94%at1000◦C.Rutile is the most stable crystalline phase of TiO2and the phase transformation(anatase to rutile)depends on both the size and dopant present in the system[26].Sui et al reported that anatase-type TiO2nanostructures transformed into rutile phase(56%)after calcination at600◦C[18],whereas only 12%of the nanomaterial was converted into rutile at800◦C in the present study.Hence,consistent with the literature for non-nanotubular structures[23],the anatase phase of the bimetallic nanotubes can be stabilized by modifying titania with a small amount of ZrO2.However,the mechanism by which the zirconia stabilizes the TiO2anatase phase at higher temperatures is unclear.As shown in the XRD data,no distinct zirconium peak was observed,indicating that zirconia was well integrated into the anatase structure.This suggests that particle agglomeration was not favored and the particles grew by the Oswald ripening process during heat treatment[26].Due to the presence of zirconia,the Oswald ripening process was restricted,reducing the crystal growth rate and increasing the phase transformation temperature.Once individual crystallites reach a threshold size,a spontaneous phase change can occur.The rapid growth of rutile particles formed during heat treatment suggests that the growth mechanism consists of particle agglomeration or grain coalescence by grain boundary diffusion[36].There may be a threshold size limit for this transformation,below which no transformation occurs,which was>27nm for this study, similar to that reported for SiO2and ZrO2doping in a titania matrix[26,36].For this reason,no rutile phase appeared up to 700◦C.When calcining at800◦C,the crystallites size became >27nm and the anatase phase started transforming rapidly into the rutile phase.In addition to the XRD data,detailed information on the structural transformations and crystal growth can be obtained using HRTEM.The HRTEM micrographs of the as-preparedFigure7.The Raman spectra of Zr–TiO2nanotubes calcined at: (a)500,(b)600,(c)700,(d)800,(e)900and(f)1000◦C.(A–anatase,R–rutile).nanotubes were amorphous(figure6(a))with no ordered structure.The lattice image of nanotubes calcined at500◦C is given infigure6(b),showing a grain size of approx.12nm width with a d spacing0.35nm,very close to the lattice spacing of the(101)planes of the anatase phase.However, all grains were not the same in terms of size and shape. Some were long with significant lattice mismatch and grain boundaries.All these defects prevent rapid grain growth.The HRTEM image for the material calcined at1000◦C is given infigure6(c),where little amorphous phase is observed.The crystallites were very large having a d spacing of0.245nm, which value is very close to the rutile(110)plane of titania. These observations also support the XRD and TEM analysis.3.4.RamanTo further verify these results,Raman spectra for the bimetallic nanotube samples were measured for several different calcination temperatures,as shown infigure7.The spectrum for the sample calcined at500◦C(figure7(a)) shows Raman peaks at142,395,517and639cm−1,which can be assigned to the E g,B1g,B1g/A1g and E g modes of the anatase phase of titania,respectively,which agrees with published values[37].With increasing calcination temperature infigures7(b)–(d),the intensity of the anatase phase increased, indicating a larger particle size being present,with the anatase peak shifting to lower frequencies.Lottici et al[38]explained this effect as the size-induced pressure effect on the vibrational modes,with the smaller the crystallite size,the higher the pressure and Raman frequencies.After calcining at900◦C (figure7(e)),three new peaks at230,442and612cm−1 appeared,which match the literature values for the rutile phase[37,38].Upon calcining at1000◦C(figure7(f)), all anatase-related peaks vanished and only the rutile-related peaks remain,indicating complete anatase to rutile phase transformation.The Raman results agree with the previous characterizationresults.Figure8.Ln of anatase and rutile crystallite size in nm as a function of the reciprocal of absolute temperature according to equation(3). (—anatase,•/◦—rutile).3.5.Activation energy of phase transformationsThe activation energy(kJ mol−1)of phase transformations can be calculated from the slope of a plot of ln rutile weight fraction versus the reciprocal of annealing temperature from XRD spectra according to Burns et al[39].This relationship is given asE a=−∂ln(X r)∂(1/T)R(3) where T is the temperature in kelvin,R is the universal gas constant(8.314J mol−1K−1)and X r is the weight fraction of the rutile phase as determined using equation(2).Figure8 shows the plot of the logarithm of the average crystallite size versus the reciprocal of the calcination temperature(solid lines),according to equation(3).A linear relationship is observed and the activation energy for the crystal growth of the Zr–TiO2nanotubes was calculated as12.8kJ mol−1and 36.4kJ mol−1for anatase and rutile phases,respectively.The activation energy for the phase transformation from anatase to rutile was also calculated,as shown by the dashed line in figure8,with a value of171kJ mol−1.These values are higher than those reported for nanocrystalline pure titania[39,40], which is a beneficial effect of doping.In explanation,during the heat treatment,there are two competitive processes:grain growth and A→R phase transformation in the nanocrystalline materials.Both processes are easier for small grain-sized materials because the activation energy for growth and phase transformation is lower[41].3.6.BET analysisThe textural properties,i.e.the surface area,pore volume and pore size distributions of the as-prepared and calcined bimetallic nanotubes were characterized by nitrogen adsorp-tion studies.Figure9shows the nitrogen adsorption isotherms for both the as-prepared and calcined materials,which exhibit H3hysteresis loops(to800◦C),typical for mesoporous materials.The isotherm for the bimetallic metal oxide nanotubes calcined at1000◦C changes to a type I isotherm, typical for a microporous material[42].The lower limit ofFigure9.N2adsorption/desorption isotherm of the Zr–TiO2 nanotubes calcined at differenttemperatures.Figure10.BJH pore size distribution of the Zr–TiO2nanotubes calcined at different temperatures.the relative pressure for the hysteresis loop is characteristic of a given adsorbate at a given temperature[43].It can be seen fromfigure9for both the as-prepared nanotubes,and for those calcined at300and400◦C that the lower pressure limit of the hysteresis loop is at P/P0=0.4.Calcinations at higher temperatures increase this value,e.g.at600◦C,P/P0increasesto approx.0.55,while after800◦C this value increases to0.6, indicating that the pores are becoming larger as the materials are calcined at higher temperatures.To evaluate the pore size distribution,the as-prepared and calcined materials were plotted as shown infigure10. The average pore diameter for the as-prepared nanotubes is approx.3nm,whereas upon calcination the pore size became gradually larger and the pore size distribution shifted,forming larger pores at the expense of the smaller ones.At600◦C the pore size is approx.9nm while,when the nanomaterials were calcined at800◦C,the pore size became more than double at approx.19nm.Calcining the materials at1000◦C collapsed the small pores,resulting in only larger pores.The surface properties of the Zr–TiO2nanotubes calcined at different temperatures are summarized infigure11.The as-prepared Zr–TiO2nanotubes,having a surface area of 430m2g−1,gradually decrease through calcination.Due to the sintering phenomena,the small pores collapse,reducingthe Figure11.Surface area and pore volume of the Zr–TiO2nanotubes calcined at different temperatures as a function of calcination temperature.pore volume and surface area.The transformations into anatase and rutile crystalline phases will also help to reduce the surface area.Interestingly,literature values show that pure titania has almost zero surface area at this high temperature[44,45]. Hence,the presence of10%zirconia increased the thermal stability and reduced grain growth rates during the course of heat treatment,resulting in moderate(23m2g−1)surface areas at very high temperature.4.ConclusionsThe thermal behavior of the Zr–TiO2nanotubes synthesized by an acid-modified sol–gel process in scCO2has been investigated in detail using SEM,TEM,TG,EDX,FTIR, XRD,HRTEM,Raman and BET analysis.SEM and TEM analysis confirmed that the morphology of the nanotube structure was preserved at up to800◦C,whereas further heat treatment deformed the tubes.FTIR and EDX analysis showed that different organic residues were removed,depending on the calcination temperature.Along with HRTEM and Raman, XRD results showed anatase nanocrystallites were formed after calcining at400◦C,while no rutile phase appeared until calcining at700◦C,with further heat treatment resulting in a rutile phase transformation and ZrTiO4being formed. The activation energy for anatase and rutile crystal growth was calculated and the values were12.8and36.4kJ mol−1, respectively.The activation energy for phase transformation was determined to be171kJ mol−1,higher than that of pure titania nanomaterials.The as-prepared nanotubes had a430m2g−1specific surface area(SSA),whereas after calcining at1000◦C the SSA was reduced to23m2g−1. Hence,the ZrO2present in the titania matrix increased thermal stability,reducing grain growth resulting in smaller crystallites, and hence preserving the morphology and surface area at high temperatures.AcknowledgmentsThe authors would like to thank Nancy Bell from the UWO Nanotechnology Centre for the SEM analysis,Ronald。

凯勒度量的里奇形式-概述说明以及解释1.引言1.1 概述凯勒度量的里奇形式是微分几何中的一个重要概念，它描述了曲率对于度量的影响。

凯勒度量是一种特殊的度量，它满足局部欧几里德性质和全局特殊正则性质。

而里奇形式是通过度量定义的一个与曲率有关的二次微分形式。

在研究从局部到全局几何结构变化的过程中，凯勒度量的里奇形式起到了至关重要的作用。

通过研究里奇形式可以揭示出度量的几何特征，进而推导出曲率的性质。

因此，理解凯勒度量的里奇形式对于研究曲率以及相关的几何问题非常重要。

在本文中，我们将对凯勒度量的定义进行介绍，它是一种满足严格的正则性质的度量。

然后，我们将引入里奇形式的定义，它是通过度量所导出的一个与曲率相关的微分形式。

最后，我们将详细讨论凯勒度量的里奇形式，并探讨它在几何学中的应用。

通过研究凯勒度量的里奇形式，我们可以深入了解曲率对于度量的影响，并从中揭示出各种几何对象的性质和结构。

这对于解决复杂的几何问题以及推动数学和物理学等学科的发展具有重要意义。

在接下来的章节中，我们将进一步阐述凯勒度量的定义和里奇形式的定义，以及它们之间的关系。

通过深入研究这些内容，我们将更加全面地理解凯勒度量的里奇形式以及其在几何学中的重要性。

1.2文章结构文章结构部分的内容可以按照以下方式编写：1.2 文章结构本文将按照以下结构进行阐述和探讨凯勒度量的里奇形式的相关内容：2.正文2.1 凯勒度量的定义首先，我们将介绍凯勒度量的基本概念和定义。

凯勒度量是一种用于度量曲面上各点之间距离的工具，它能够衡量曲面的局部几何特征。

我们将详细解释凯勒度量的数学定义和性质，阐述其在几何学和物理学中的应用。

2.2 里奇形式的定义接下来，我们将引入里奇形式的概念和定义。

里奇形式是一种在黎曼流形上定义的双线性对称张量，它起着衡量黎曼流形曲率的作用。

我们将讨论里奇形式的数学定义、性质和作用，并介绍其在物理学中的应用。

2.3 凯勒度量的里奇形式在这一部分，我们将探讨凯勒度量与里奇形式的关系。

Mathematical ProblemsLecture delivered before the International Congress ofMathematicians at Paris in 1900By Professor David Hilbert1Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the "problem of the line of quickest descent," proposed by John Bernoulli. Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermat, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted, as is well known, that the diophantine equationx n + y n = z n(x, y and z integers) is unsolvable—except in certain self evident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors—a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very different region of research, I remind you of the problem of three bodies. The fruitful methods and the far-reaching principles which Poincaré has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.In order to throw light on the importance of certain problems, I may also refer to Weierstrass, who spokeof it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi's problem of inversion on which to work.Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics. But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. Thus arose the problem of prime numbers and the other problems of number theory, Galois's theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions; indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in this ever-recurring interplay between thought and experience.It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning. Indeed the requirement of rigor, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and, on the other hand, only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially when it comes from the world of outer experience, is like a young twig, which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem, the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor. Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigorous function-theoretical methods and the consistent introduction of transcendental devices. Further, the proof that the power series permits the application of the four elementary arithmetical operations as well as the term by term differentiation and integration, and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis, particularly of the theory of elimination and the theory of differential equations, and also of the existence proofs demanded in those theories. But the most striking example for my statement is the calculus of variations. The treatment of the first and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations. By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of the calculus of variations. For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum, the calculation of the second variation and in part, indeed, the wearisome reasoning connected with the first variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment. This opinion, occasionally advocated by eminent men, I consider entirely erroneous. Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation? Whocould dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content. Just as in adding two numbers, one must place the digits under each other in the right order, so that only the rules of calculation, i. e., the axioms of arithmetic, determine the correct use of the digits, so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen.2Some remarks upon the difficulties which mathematical problems may offer, and the means of surmounting them, may be in place here.If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems. The introduction of complex paths of integration by Cauchy and of the notion of the IDEALS in number theory by Kummer may serve as examples. This way for finding general methods is certainly the most practicable and the most certain; for he who seeks for methods without having a definite problem in mind seeks for the most part in vain.In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them bymeans of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and it seems to me that it is used almost always, though perhaps unconsciously.Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant C, or the existence of an infinite number of prime numbers of the form 2n + 1. However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility. I instance the problem of perpetual motion. After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;3 and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry. The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of thecontinuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky. I therefore first direct your attention to some problems belonging to these fields.1. Cantor's problem of the cardinal number of the continuumTwo systems, i. e, two assemblages of ordinary real numbers or points, are said to be (according to Cantor) equivalent or of equal cardinal number, if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other. The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving. This is the theorem:Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3,... or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line; as regards equivalence there are, therefore, only two assemblages of numbers, the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage; the proof of this theorem would, therefore, form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before b and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. But there are, as is easily seen infinitely many other ways in which the numbers of a system may be arranged.If we think of a definite arrangement of numbers and select from them a particular system of these numbers, a so-called partial system or assemblage, this partial system will also prove to be ordered. Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way, that not only in the assemblage itself but also in every partial assemblage there exists a first number. The system of integers 1, 2, 3, ... in their natural order is evidently a well ordered assemblage. On the other hand the system of all real numbers, i. e., the continuum in its natural order, is evidently not well ordered. For, if we think of the points of a segment of a straight line, with its initial point excluded, as our partial assemblage, it will have no first element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have a first element, i. e., whether the continuum cannot be consideredas a well ordered assemblage—a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out.2. The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms. The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them4 and in so doing replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows: that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.To show the significance of the problem from another point of view, I add the following observation: If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -l does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved. In the case before us, wherewe are concerned with the axioms of real numbers in arithmetic, the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum. Indeed, when the proof for the compatibility of the axioms shall be fully accomplished, the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless. The totality of real numbers, i. e., the continuum according to the point of view just indicated, is not the totality of all possible series in decimal fractions, or of all possible laws according to which the elements of a fundamental sequence may proceed. It is rather a system of things whose mutual relations are governed by the axioms set up and for which all propositions, and only those, are true which can be derived from the axioms by a finite number of logical processes. In my opinion, the concept of the continuum is strictly logically tenable in this sense only. It seems to me, indeed, that this corresponds best also to what experience and intuition tell us. The concept of the continuum or even that of the system of all functions exists, then, in exactly the same sense as the system of integral, rational numbers, for example, or as Cantor's higher classes of numbers and cardinal numbers. For I am convinced that the existence of the latter, just as that of the continuum, can be proved in the sense I have described; unlike the system of all cardinal numbers or of all Cantor s alephs, for which, as may be shown, a system of axioms, compatible in my sense, cannot be set up. Either of these systems is, therefore, according to my terminology, mathematically non-existent.From the field of the foundations of geometry I should like to mention the following problem:3. The equality of two volumes of two tetrahedra of equal bases and equal altitudesIn two letters to Gerling, Gauss5 expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.6 Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.74. Problem of the straight line as the shortest distance between two pointsAnother problem relating to the foundations of geometry is this: If from among the axioms necessary to。

A CONSTRAINT MONITOR ALGORITHMFOR THEGlennand Robert D.4800 Oak Grove Drive Pasadena, CA 91109-8099AbstractTheConstraint Monitor algorithm performs appropriate checks to ensure the legality and the realizability of the instantaneous attitude, rate, and acceleration commands.When the command is found to be unrealizable becauseof hardware limitations ortionand Science ObjectivesThecarrying a probe intended for delivery into the Titan atmosphere. The probe entry into the Titan atmospherewill occur about four months after Saturn arrival.Senior Technical Staff, Guidance and Control Analysis Group, Automation and Control Section,Member AIAASaturn’s atmospheric composition,winds and temperature,configuration and dynamics of the magnetosphere, structure and composition of the rings,characterization of the icy satellites, and Titan’s atmospheric constituent abundance. The rack mapperwill perform surface imaging and altimetry during eachTitan flyby.Flyby) andconduct scientific investigations for 120days andthe cancellation ofspacecraft. Both the high and lowprecision scan platforms and their structural booms weredeleted,the role of the observation platform and the entire spacecraft now had to turn in order to do earth pointing,star tracking, and remote science pointing. This makes the detection and avoidance of spacecraft attitudeconstraints ever more difficult since movement of oneand Modelingspacecraft with [heboom and three radio and plasma wave science antennas. The core structure of the spacecrafthouses the propulsion module which consists of twotwo1American Institute of AeronauticsFigure 1.axis. Towards theend of the Saturn tour (probe released), the mass and inertia will be reduced to 2400 kg and 6940, 5720, 3600kg-m 2. The magnetometer boom has a fundamentalfrequency at 0.65The three radio and plasma wave antennas arc much lower in mass and inertia, and have a frequency of 0.13Hz or higher and a damping of 0.2 ‘%.At the beginning of the mission, the spacecraftpropellant mass is heavier than the spacecraft dryandNitrogenconsumed by the mainengine, has a total launch mass of 3000 kg. The twolaunch mass isaxis. Each of the twoArticulation Control Subsystemspacecraft burn maneuvers, constraint violations detection and avoidance, fault protection, command processing, telemetry generation and data handling. The attitude control functional block diagram is shown inFigure 2.Diagram2American Institutenever violate a constraint regionif the ground command sequence is generateda clear understanding of the two stepsinvolved in the process -- the detection of constraint violations and, once a violation has been detected, the steps involved in the avoidance process. The avoidance step is where an acceptable alternate path is generated,In the following,underlined capital letters denotematrices, underlined()’implies transposition. The subscript b is used to denotetocoordinates.The symbol ‘x’ implies vectorcross product. Atworepresent thevector component,MonitorThe Constraint Monitorobjects, implements part ofthe AACS fault protection functions.It determines what commanded motions are acceptable, when action should bc taken to interrupt the ACM commanded pointing profile so that an unacceptable motion is notrealized, andwith the ACMcommanded motion, should the nominal path becomesmooth in attitude and rate, withintolerances (rate limited) and realizable (acceleration limited).The two functions performed by the algorithm mayThe Detection function provides the capability to detect constraint violations for ACM-commanded motions.The algorithm is capable of detecting imminent violations so that action may bc taken well in advance. The Avoidance function generates the alternate path which satisfies all constraints when ACM commanded path has been found to be in violation. The computational needs of the Avoidance algorithm are significantly greater than that of the Detection algorithm.In order to make CMT run in real-time,only the Detection algorithm isThe avoidance algorithm cm the otherhand executes in the background at a slower pace (onceevery 2 seconds). Thekinematic extrapolations, carried out every 125 rnsec,which usc the last commanded attitude and rate, and the by CMT may beclassand dynamic.Flight rulesrestricting the pointing of spacecraftcelestial objects are translated into celestial constraints enforced by the CMT. Enforcement of these constraints protect certain spacecraft fixed directions from exposure to celestial objects. There are two types of celestial constraints - non-timed and timed. A non-timedconstraint may be expressedA timed constraint, which protects certain S/C axes from extended exposure to celestial objects, may be expressed as :qThe angular separation between body vector Bfor atime period greater than T. Time is accumulated atthe rate of a 1a minimum of zero.Celestial constraints define a conical region around the celestial vector. The body vector is required to stay out of the cone for non-timed constraints and for timed constraints it is required that the body vector not stay inside the cone for longer than the specified duration.CMT also enforces dynamic constraints to ensure that the acceleration and rates commanded by the ACM are not excessive. These constraints are specified as rate and acceleration ellipsoids in the spacecraft fixed coordinates.The sizes of these ellipsoids can becontrolled via ground commands.The celestial constraints enforced by CMT are defined by ground commands and stored in a table on board (Table 1). The table is of a fixed size (20 entries) and stores all relevant attributes of celestial constraints,which are, the constraint name (unique), the name of the body object, the name of the inertial object, the spatial extent (the half-cone angle of the constraint cone ), the temporal extent (timed constraints only - the constraint is treated as a hard constraint when this attribute iszero),turned OFF (neither DETECT nor AVOID)The celestial and the body vector identifiers (C and B,respectively, in the examples above) and measuresThe appropriate vector valuesTwoseparate tables inthe appropriate vector value is returned by IVP.Once a constraint violation has been detected, theAvoidance function allows the ACM-computed attitude,rate and acceleration to be modified such that theresulting attitude,rate and accelerationAvoidance machinery does not engage unless CMT is currently in Avoidance and / or an AVOIDable celestial constraint violation or a dynamic constraint violation has been detected.When “DETECT only” celestial constraints are violated, the ACM computed attitude is not altered but such a violation is reported. The CMToutput is00AVOIDDROPSUNB37075I .0OFFDROP.1. Constraint TableDetectioncelestial constraint and dynamic constraint violations.Two types of constraints are covered:celestial constraint violations. We would like to start taking evasive action early enough so that the constraint cone is not entered for non-timed constraints and not entered for more than the allowed time for timed-constraints. This requires extrapolation of the current is lessthan the sumstoppingdistance is the predicted angular motion in the direction of the constraint under the assumption that a radialacceleration is applied away from itA closed-form solutionfor this distance is not possible. Instead a first order approximation is used. It works very well, especially for the relatively weak acceleration and rate capabilitiesofrespectively) providesthe direction (henceforth referred to as the “escape”direction) used in the evaluation of the stoppingdistance. Let theisthe normalization operator andused in this evaluation. TheAvoidance function applies maximum availableacceleration in theor is imminent (it isnot necessarily time-optimal; it would be, if all axeshad theIf the body vector is moving away from the celestialvector (else it is computed as the angular distance moved such that the application of maximum decelerationabout The prediction requires the usc of maximum available acceleration along the escape axis. The use of largest available acceleration might be optimistic since, while trying to escape, the escape direction may change as the offending body vector is pushed away from the celestial constraintit is trying to avoid. To avoid this optimism (and therefore future violations) and err on the conservativeside, the smallest available acceleration is usedused in constraint avoidance.is therefore evaluated as0.5A violation is declared imminent if the predicted future c-bseparation+is smaller than the constraintangleinside the constraint cone, the time must be accumulated. If the body vector is outside the constraint cone, the accumulated time must be reduced at the rate R specified in the constraint definition. In other wordstwhere A is the time elapsed since the last evaluation.This accumulation is bounded by O from belowentered as the S/Ca violation. If the constraint ispredicted toexited in the time5American Institute of Aeronautics and Astronauticsremaining ( acceleration then there can not be a violation imminent,otherwise an imminent violation isAvoidance machinery is invoked.acceleration commands must never lie outside theirrespective ellipsoids. A dynamic constraint violation isacceleration constraints imposed by theimminent by the Detection function.in compliance of all constraints andwhich will prevent the S/C attitude from violating celestial constraints and the rate and accelerationcommands from exceeding theDetection cycles). On initial entry intois accounted for in the avoidanceacceleration evaluation.The corrective action taken is designed to be such that the constraint in question is not violated eventually.The corrective action consists of three steps, which maybe repeated several times until the CMT commanded motion merges with the ACM commanded motion.The end goal is to always try to move constraints and is close to the ACM commanded path.When the ACM is not in violation, the goal is of course the ACM command itself. The first action takenwhen a celestial constraint violation isno longer in imminentviolation. If at this point it is possible to move along a great circle towards the goal attitude and not violate any constraint in the immediate future (next 2 seconds),aconstraints arc allowed to be penetrated (under certain conditions) but non-timed constraints arc generally followed at a fixed distance. This behavior is referred to as “... following the flow-field”. While in Circulatemode, the attitude is continually checkedfor imminent violations (transition back toEscape) or possibility ofcelestial constraints. When there are no constraint violations, the attitude and rate goals coincide with the ACM commands. When one or more celestial constraint violations arc present, the goal attitude issuch that all celestial constraints (whichhas been assumed since the attitude goal canAstronauticsexhibitare an artifact of trying tocompute the attitude goal in a finite number of steps, It makes no sense therefore to compute a rate goal consistent with the variation in the attitude goal. Only the violated constraints are considered in determining the attitude goal. The attitude goal is not “close” to the ACM command necessarily but it does satisfy all violated constraints (only in rare cases have we found the algorithm to yield a non-compliant attitude: rapidlymoving celestial constraintsfrom the goal it is tryingto attain, largest possible rate is commanded until it is in the vicinity of it where a rate proportional to thedistance from the goal is commanded.It was noted earlier that while in the Circulate mode,the constraints have to bethen i = 1, 2, 3 are theRodriguez parameters) - referred henceforth as theathe “attitude”, is required to stay outsidea collection of hyperboloids.In general the attitudevectorbe the constraint celestial vector in inertialcoordinates,S/Cw=wherex 3 identity matrix. Attitude has to “flow” a certain way around a collection of such surfacesasaxis. This argument sets the flow-field direction in a relative sense. The correct orientation (i.e. one way orthe other) isAttitudeWhen CMT is in avoidance the attitude does not wander aimlessly, it is actually trying to move towards an attitude goal as it negotiates the constraints betweenthe current attitude and the attitude goal. When the celestial constraints, the goal attitude and rate are the ACM-commanded values. The goal attitude and rate are not close, necessarily, to the ACM attitude (itis, inThe algorithm starts by accepting the ACM attitudecelestial constraintviolation (by the candidate goal attitude that is), thecandidate goal attitude is updated such (hat the offending body vector (at the modified goal) lies slightly outside the edge of the constraint cone. The movement of the offending body vector is accomplished by rotating thebody vector about the escape axisrepeated until there are no violations during an iterationor the iteration count exceeds the number of constraintsunder consideration (whichever comes first).commanded attitude.7American Institute of Aeronautics and Astronautics3.2.3.Desired Direction of MotionOnce a goal attitude is available, the direction in which CMT should try to head towards is needed. The direction is a function of’ the current state, the goal state (attitude and rate), and the location of the constraints which are to be avoided. The desired direction of motion is prescribed in terms of a rate vector which CMT should try to attain by the end of next Avoidance cycle except in Escape mode, where the desired direction is actually a direction in which to accelerate in.When in the Escape mode, the desired direction of travel is the weighted sum of all applicable escape directions and a maximum possible acceleration is commanded in this direction until either the constraints are no longer in imminent violation or the rate limit is reached.When the constraints have been exited (i.e. an Escape is no longer needed), a transition to either the Circulate or the Clear mode takes place. These modes were describedearlier. A Clear mode isbecomes a linear function of the separation from the goal.(1)+ KwherecoordinatesandThe ()* is theconjugation operation. Note that only the vector part oftheimminent had CMT tried to move towards the goal. A transition to Circulate mode is then declared until it is possible to move towards the goal (Clear mode). A desired rate vector has to be computed again. The desired rate magnitude is the largest possible when far from the goal, else the magnitude is the norm ofsum of the so-called “circulation vectors”. The weights bear an inverse square relationship to thedistance from the edge of the constraints.aboutaboutis small. Tbc attitude perturbation+where s =Clearly, aswhich implies a rotation about the directionthat is left to do is to compute therequired acceleration command. Evaluations are differentdepending onLargest possible acceleration is applied in this directionuntil the mode can be transitioned to Circulate or Clearor maximum rate is reached.For non-Escape situations, the desired rate vector (1) isused to compute the acceleration command:is the prescribed rate (1) and CMT-commanded rate in S/C coordinates. It can beshown that, inacceleration, results in a stable behavior when0<It ismust be preserved if possible. A geometric solution ispossibleis shortened such that the accelerationcommand lies on the acceleration3.2.4. Avoidancecommanded path is not in path (in attitude and rate) is close to the ACM-commanded path. The maximum rategroundbeforeattitude goal. Protection against such occurrences isprovided by another attribute of the constraints (the DROP/KEEP column in Table 1). When a situation such as the one described here arises (i.e. away from the goal attitude for too-long) all non-Sun breed constraints (i.e. constraints for which the inertial vector is not the S/C to Sun direction) and all Sun-based constraints whose DROP/KEEP attribute is DROP are marked DETECT only (i.e. the type is changed to DETECT) by CMT. No ground intervention is needed in this step.The remaining constraints which have to be negotiated are therefore all Sun-based and it can beshown that The actions of the algorithm are demonstrated below viasomeexamples. First consider a simplekept out of it isthe spacecraft x-axis. First consider a case where ACM commands a path which moves the spacecraft x-axis,initially outside, through the constraint to a final point outside the constraint.The two avoidanceexamples for this ACM command arc shown in the top two sub-plots in Figure 3. The intersections of the spacecraft x-axis with the unit celestialspherex-axis path goes from right to left and is shown as the thin solid line and the CMT action appears as a dot plotted every 2 seconds.The bard constraint is never entered and CMT rejoins with ACM after an Escape-Circulate-Clear sequence.The timeconstraint (top right) is indeed entered but for only(the bottomplots), CMT seeks the goal attitude, just outside the constraint but close to the ACM point, and comes to rest there. Again, the hard constraint is never entered but the timed constraint is entered but exited in time (0.875 seconds to spare). Note that repeated attempts arc made by CMT to approach the ACM commanded location inside the timed-constraint. This is of courseperfectly legal as long as the06 —–.03Next we consider a slightly more complicated examplewhere three constraints have to be enforced. Theconstraint table is as shown in Table 2. Three celestialNameCelestial BedyHalf Cone Max Allowable Decay Rate TypeDrop/Object, C Object, BAngle, Time, T (see)R,AVOIDKEEP 450AVOID KEEPcAVOIDKEEPAmerican Institute of Aeronautics and Astronauticsconstraints are to be enforced.The first two (A, B)). Componentsof various vectors used to define constraints A, B, Ccoordinates).Table 3. Vector Components for Constraints in Table 2The constraint cone projections and projections ofvarious axes on the unit J2000 celestial sphere arcshown in Figure 4. Constraint regions are regionsinside the dashed circles labeled A, B, C representing thethree constraints A, B, C, respectively. Theand05.e420,c,..,Figure 5. Attitude Separation and CMT ModesSummary1This the design of an autonomousboard theperformance has been very satisfying indeed.6.AcronymsThe work described in this paper was carried out by theJet Propulsion Laboratory, California Institute ofTechnology,under a contract with the NationalAeronautics and Space Administration.8. References[1]Rasmussen,Guidance and Control Conference, Keystone,Colorado, February 1-5, 199510American Institute of AeronauticsAmerican Institute of Aeronautics and Astronautics。