Stability and quasinormal modes of the massive scalar field around Kerr black holes

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2022年考研考博-考博英语-合肥工业大学考试全真模拟易错、难点剖析AB卷(带答案)试题号：10

2022年考研考博-考博英语-合肥工业大学考试全真模拟易错、难点剖析AB卷（带答案）一.综合题(共15题)1.单选题The energetic ultraviolet radiation is ______ to life since it tends to break up the complicated molecules on which life depends.问题1选项A.indissolubleB.indolentC.indiscreetD.inimical【答案】D【解析】考查形容词辨析。

A项indissoluble“不能分解的，不能溶解的”，B项indolent“懒惰的，无痛的”，C项indiscreet“轻率的，不慎重的”，D项inimical“敌意的，有害的”。

由原因状语since it tends to break up the complicated molecules on which life depends“因为它倾向于分解生命所依赖的复杂分子”可知，“有害的”符合语境。

句意：高能的紫外线辐射对生命是有害的，因为它往往会破坏生命所依赖的复杂分子。

因此，该题选择D项正确。

2.填空题For several years, scientists have been testing a substance called interferon(干扰素)，a 36 wonder drug that is proving to be effective in treating a variety of ailments, including virus infections, bacterial infections, and tumors. To 37 , the new drug has provoked no negative reaction of sufficient significance to 38 its use. But in spite of its success, last year only one gram was produced in the entire world.The reason for the 39 lies in the structure of interferon. A species of specific protein, the interferon produced from one animal species cannot be used in 40 another animal species. In other words, to treat human beings, only interferon produced by human beings may be used. The drug is produced by infecting white blood cells with a virus. Fortunately, it is so 41 that the amount given each patient per injection is very small.Unlike antibiotics, interferon does not attack germs directly. 42 , it makes unaffected cells resistant to infection, and 43 the multiplication of viruses within cells.As you might conclude, one of the most 44 uses of interferon has been in the treatment of cancer. Dr. Hans Strander, research physician at Sweden’s famous Karolinska Institute, has treated more than one hundred cancer patients with the new drug. 45 a group of selected patients who had 46 surgical procedures for advanced cancer, half were given conventional treatments and the other half were given interferon. The 47 rate over a three-year period was 70 percent among those who were treated with interferon as 48 with only 10 to 30 percent among those who has received the 49 treatments.In the United States, a large-scale project supported by the American Cancer Society is now under 50 . If the experiment is successful, interferon could become one of the greatest medical discoveries of our time.36. A. possible B. adequate C. potential D. capable37. A. time B. stage C. date D. period38. A. determine B. discourage C. decrease D. disclose39. A. uniqueness B. peculiarity C. feature D. scarcity40. A. treating B. producing C. operating D. handling41. A. potent B. slight C. obvious D. obscure42. A. However B. Moreover C. Hence D. Instead43. A. motivates B. prevents C. strikes D. eliminates44. A. restrictive B. dramatic C. probable D. crucial45. A. With B. For C. Among D. Within46. A. sustained B. endured C. prepared D. undergone47. A. existence B. treatment C. survival D. growth48. A. weighed B. balanced C. differentiated D. compared49. A. original B. conventional C. exceptional D. desirable50. A. way B. track C. course D. route【答案】36.A37.C38.B39.D40.A41.A42.D43.B44.D45.C46.D47.C48.D49.B50.ANormal 07.8 磅 02false false falseEN-US ZH-CN X-NONE 【解析】36.考查形容词辨析。

生化出题092

生化出题生物092 酶、维生素3.The k cat of carbonicanhydrase is 106 s-1,while the k cat of catalase is 107s-1. Which is more efficient in catalyzing reactions?_____A.Carbonic anhydrase.B. Catalase.C. There is not enough information to make a conclusion.D. They are same.4.Which reaction is not multisubstrate reaction? _____A.Random reactionB.Ordered reactionC.Ping-pang reactionpetitive reaction5.Which describe about ping-pang reaction is right? _____A. The altered enzyme is restored to its original form before second product is bound.B. The altered enzyme is restored to its original form before the first product is released.C. The first product is released after which the second substrate is bound.D. The first product is released before which the second substrate is bound.6. Which description about sequential reaction is not right?_____A.ordered reaction is a kind of sequential reaction.B. random reaction is a kind of sequential reaction.C. sequential reaction requires all the substrates to be present before any product is released.D.sequential reaction requires all the substrates to be present after any product is released.7.When a competitive inhibitor is bound to an enzyme molecule,how will V max and K m change?_____A. V max remains unchanged, K m increasesB. V max decreases,K m decreasesC. V max remains unchanged, K m decreasesD. V max decreases,K m increases8. Which of the following is wrong?_____A. Reversible enzyme inhibition includes competitive inhibition、uncompetitive inhibition、noncompetitive inhibitionB. Dialysis method can identify reversible inhibitionC. Cofactor or coenzyme can contribute to a variety of enzyme activityD. Remove the cofactor or coenzyme the enzyme will be denatured protein inactivation9. Which is the characteristic of competitive inhibition?_____A. Inhibitors have a similar structure with the substrateB. The combination of inhibitor binding does not affect the substrateC. V max decreasesD. K m remains unchanged10. Which statement about isoenzyme is right?_____A. They have the same catalytic reactionB. They have the same molecular structureC. They have the same immunological propertiesD. They have the same kinetic parameters11. Which statement about allosteric enzyme is wrong?_____A. Allosteric enzymes can change the activity of enzymesB. Allosteric enzymes can be combined with the catalytic part ofenzymesC. Allosteric enzymes can reversible binding with the enzyme moleculesD. Allosteric enzymes can change the conformation of the enzymes12. Zymogen is inactive becauseA. Active center is not formed or exposedB. Zymogen is nomal proteinC. Lack of coenzymes or cofactorsD. Zymogen is denatured protein13. Organophosphorus pesticides as inhibitors of the enzyme is acting on _____part of the enzyme active center.A. SulfhydrylB. HydroxylC. CarboxylD. Imidazole14. What is the physiological meaning of zymogen activation?_____A. Speed up the metabolism.B. Restore the activity of enzymes.C. The way of biological self-protection.D. The way of protecting enzymes.15. Zymogen is the precursor of enzymes _____A. It has activity.B.I t hasn’t activity.C. It can increase activity.D. It can decrease activity.16. Allosteric enzyme is_____A. Monomeric enzymeB. Oligomeric enzymeC. Multienzyme complexD. Michaelis enzyme17.The most recognized theory about enzyme and substrate binding is?_____A. Theory of the active centerB. Induced fit theoryC. Lock-key theoryD. Intermediate theory18.Many chemical reactions have ionic intermediates. There are two types of ionic intermediates：One species is electron-rich ，or____B___, and the other species is electron-poor , or ____A___.A.ElectrophilicB.NucleophilicC.Cleavage reactionD.Oxidation-reduction reactions19.Another type of nucleophilic substitution involves direct displacement. In this mechanism ,the attacking group or molecule adds to the face of the central atom opposite the leaving group to form a ________having five groups associated with the central atom.A. Oxidation stateB. Reduction stateC. Transition stateD. Equilibrium state20.This is an example of a ___________ reaction.A .Nucleophilic substitutionB .Electrophilic substitutionC .OxidationD. Reduction21 .Which effects will happen in enzymatic reaction?_____A. Raise energy levels of products.B. Reduce activation energy of the reactions.C. Raise activation energy needed to start reactions.D. Reduce energy levels of reactants.22. Which of the following ones supports induced-fit theory?_____A. The relationship between enzyme and substrate is same as it between lock and key.B. Enzyme active center is variable, only when the spatial structure of it changes under the influence of substrate, could reactions happen.C. Conformation of the enzyme does not change but the structure of substrate changes for adapting into enzyme active center.D. Substrate analogues cannot induce the change of enzyme’s conformation.23. The correct statement of enzyme inhibitor is_____A. Enzyme inhibitors are enzyme denaturants.B. Enzyme denaturants only can combine with the genes of enzyme active center.C.All the enzyme denaturants can reduce the speed of enzymatic reaction.D. Generally, enzyme inhibitors are macromolecules.24.Which two modes of enzymatic catalysis invoveionizable side chains? (Which two modes are chemical modes of enzymatic catalysis?) _____1.induced-fit2.acid-base catalysis3.proximity effect4.covalent catalysisA.1 2B.3 4C.1 3D.2 425.Which sentence is uncorrect about covalent catalysis?_____A. A substrates is bound covalently to the enzyme to form a reactive intermediate.B. Ionizable side chains participate in this kinds of catalysis.C. The acceleration of a reaction is achieved by catalytic transfer of a proton.D. The reacting side chain of the enzyme can be either a nucleophile or an electrophile.26.Which sentence about chemical mode of enzymatic catalysis is wrong?_____A. Enzymes that employ acid-base catalysis have amino acid side chains that can donate and accept protons.B. The major modes of enzymatic catalysis are acid-base catalysis and induct-fit.C. The effects of pH on the rate of an enzymatic reaction can suggest which residues paticipate in catalysis.D. Inidazoliums(咪唑基) on histidine are the most active group in acid-base catalysis.27. Which step is the rate-determining step of a typical enzyme catalyzed reaction? _____A. Binding of substratesB. Chemical catalysisC. Both A&BD. Neither A nor B28.Which type of chemical reaction CANNOT proceed a higher rate?___A. Association reaction.B. Proton transfers.C. Electron transfers.D. Covalent binding reaction.29.Superoxide Dismutase has a rate exceeds the rate of diffusion reaction because of _____A. Binding of the anionic substrate.B. Transfer of electron & proton.C. Electrostatic effect.D. Release of uncharged products.30.What determine the substrate specificity of reactions? _____A. The chemical properties of an enzyme and the shape of the amino acid residues.B. The chemical properties of the amino acid residues and the shape of the active site of an enzyme.C. The temperature of reactions.D. The solution composition.31.Which one below does not influence the degree of binding of substrates to enzymes?_____A. Mechanic reasoning.B. Measurements of the tightness of binding of substrates.C. Inhibitors to enzymes.D. The temperature of reactions.32.What an enzyme must be complementary to the transition state ?__A. In shape and chemical characterB. In sizeC. In molecular polarityD. In molecular charging33. Which one is correct in the following statement?_____A .The response in solution between two reactants is intramolecular reaction.B .In a system,if the confusion degrees is higher,the entropy would be lower.C .The intramolecularreaction involves true catalysis.D .In the intramolecular reactions, the substrate was covalently attached to the catalyst, and there was no recycling of the catalyst.34. Which one has two bonds that allow rotational freedom?____A. Glutarate ester.B. Succinate ester.C. The rigid bicyclic compound 4.D. None.35. In the compound,the interaction between enzymes and the object including_____A. Hydrogen bond \Ion key\van der Waals force \hydrophobicityB. Hydrophobicity\Ion key\van derC. Van der Waals force\ Hydrogen bond\ hydrophobicityD. Ion key\van der Waals force \hydrophobicity36. Which of the following is wrong .A. Induced-fit is not a catalytic mode but primarily a substrate specificity effect.B.Enzymes are rigid templates that accepted only certain substrates as keys.C.The transition state is a stable high-energy state.D.An enzyme is most effective if it is in the active form initially.37. What is not the factor to affect the efficiency of enzymatic catalysis?_A. Induced fit.B. Covalent catalysis.C. The proximity effect.D. Active center.38. Which of the following is electrophilic base ? .A. ImidazolylB. HydrosulphonylC. Metallic ionD. Hydroxyl39. The experiment that examines the mechanism of chymotrypsin should be in ____ environmentA. A hydrophobic environment.B. A hydrophilic environment.C. An acid environment.D. An alkalic environment.40. A proposed detailed mechanism for chymotryspin includes _____kind(s) of chemical catalysis?A. Covalent catalysisB. Acid-base catalysisC. BothD. Neither41. How to assess the relative importance of the amino acids in the Ser-His-Asp catalytic triad in the chymotrypsin and other serine proteases?_____A. Site-directed mutagenesisB. X-ray crystallographyC. Covalent modificationD. Allosteric regulation42. The serine proteases typsin, chymotrypsin, and elastase catalyze much of the digestion of peptides in the _____A. Mouth cavityB. StomachC. IntestineD. All of above43. The serine proteases are initially synthesized and stored in the pancreas as_____called zymogen.A. Active precursorsB. Inactive precursorsC. Selective proteolysisD. Serine residue44. Experiments with specifically mutated trypsin indicate that the ____at the base of its specificity pocket is a major factor in substrate specificity.A. Asp residueB. Glu residueC. Ser residueD. Met residue45. Serine proteases are a class of enzymes that _____in the protein.A. Cleave the disulfide bondsB. Cleave the peptide bondC. Both a and bD. Neither a nor b46.Cosubstrates——one type of coenzymes,are actually substrates inenzyme-catalyzed reactions, which of the followings about it is false？__A. The cosubstrate is part of active site.B.It is recycled repeatedly within the cell.C.Coenzymes shuttle mobile metabolic groups among different enzymes-catalyzed reactions.D.It remains bound to the enzyme during the course of the reaction.47. Metalloenzyme is a kind of enzyme that requires metallic cations to achieve full catalytic activity, which of the followings about it is false？_A. It contains firmly bound metal ions at their active site.B. The ions of some metalloenzymes can act as eletrophilic catalysis.C. The ions of some metalloenzymes can undergo reversible oxidation and reduction.D.Metal-actived enzymes belong to metalloenzymes.48. Cofactors can be classified into_____A. Metalloenzymes and coenzymesB. Essential ions and metalloenzymesC. Prosthetic groups and essential ionsD. Prosthetic groups and coenzymes49. When one ATP hydrolyzes completely,we can get _____A. One ribose,one adenine,three phosphoric acids.B. One deoxyribose, one adenine,three phosphoric acids.C. One ribose,one adenosine,three phosphoric acids.D. One deoxyribose, one adenosine,three phosphoric acids.50. NAD+ and NADH are the coenzymes of_____A.DehydrogenasesB. AcyltransferaseC. TransaminaseD. Decarboxylase51. The role of NAD+ and NADH is to transfer_____A. Two hydrogen atomsB. One hydride ionC.AminoD. Acyl52. Where are the FAD and FMN derived from?_____A. Vitamin B1B. Vitamin B2C. Vitamin B3D. Vitamin B553. What’s the reactive center of CoA?_____A. —SHB. —OHC. —CH3D. —NH254. Which is involved in acyl-group-transfer reactions in which simple carboxylicacids and fatty acids are the mobile metabolic groups?_____A、TPPB、FAD / FADH2C、CoAD、FAD / FADH255.Which vitamin is the composition of TPP ?_____A.B1B.B2C.B3D.B456. Pyridoxal phosphate is a kind of coenzyme formed by?_____A.B3B.B4C.B5D.B657.Which radical is biotin catalyzed and transported？_____A. HydroxideB. CarboxylC. AldehydeD. Ketone58. Tetrahydrofolate is formed from folate by adding hydrogen to positions of the pterin ring system except_____A. 5B. 6C. 7D. 959. Tetrahydrofolate is required by enzymes that catalyze biochemical transfers of how many carbon units?_____A. 1B. 2C. 3D. none of above60. Cobalamin is required as a micronutrient by the following organisms except_____A. bacteriaB. algaeC. all animalsD. plants61.In order to well absorb Vitanmin A by intestine from carrots, you'd better to cook it by _____ .A. AldehydeB. Carboxylic acidC. OilD. Water62.Which of following is the most slowly diffused agent in the silica gel column chromatography with weak polar mobile phase?_____A.Vitamin CB.Vitamin DC.Vitamin ED.Vitamin K63.Which of following agents is the strongest reducing lipid vitamins in biological pathways?_____A.Vitamin AB.Vitamin CC.Vitamin ED. Vitamin K64. Which one plays the major role in membrane-associated electron transport?_____A.ATPB.FADH2C. Coenzyme QD.NADH65. Which description of protein coenzyme is false?_____A. They do not catalyze reactions by themselves.B. They are generally less heat-stable than most enzymes.C. They can only participate in reactions with certain enzymes.D. They are smaller than enzymes.66. Which of the following names is not one of the categorizes enzymes classified according to the general class of organic chemical reaction that is catalyzed?_____A. UreaseB. TransferaseC. HydrolaseD.Ligase67. Which kind of enzyme catalyzes the reaction requiring the presence of coenzymes?_____A. KianaseB. OxidoreducateC. LyaseD. Transferase68. Which of the following statement is incorrect?_____A. In group-transfer reactions, substrates and enzymes are connected with covalent binds.B. Hydrolases are a special class of transferases with water serving as the acceptor of the group transferred.C.The enzyme catalyzing the interconversion of L-alanine and D-alanine is one of Isomerases.D. Since the enzymes catalyze both forward and reverse reactions, two-way arrows are used even when then equilibrium favors a great excess of producy over substrate.69. Which of the following coenzymes are the coenzyme of decarboxylase?_____A.NAD+B.FADC.NADHD.TPP70. Please choose the wrong statement._____A.NAD+ is the oxidation state of NADH.B.FMN/FMNH2 can transfer two hydrogen atoms.C.NADP+/NADPH are the coenzymes of dehydrogenase.D.CoA is the coenzyme of aminotransferase.71. CoQcan transfer_____A. Hydrogen atomB. ElectronC. Hydrogen atom and electronD. Hydrogen ion72.The enzymatic reaction _____A. Can decrease ΔG0B. Can increase the energy of ESC. Cannot change ΔG0 but change the rate of product and substrate.D.Accelerate the velocity of both forward reaction and backward reaction73. Which of the following demonstrations about the enzyme-substrate complex is not true? _____A. The stabilize of enzyme will increase when enzyme binds substrate.B. When the concentration of substrate reaches a certain high level, the catalytic center of an enzyme will be saturated and the velocity will be maximal.C. Due to the rapid response of substrate, it is unable to separate the ES.D. ES can be analysed by the X-ray crystal diffraction.74. Which of the following groups of enzyme is not included in the six categories which is maintained by the IUBMB(the International Union of Biochemistry and Molecular Biology )?_____A. OxidoreductasesB. TransferasesC. LyasesD. Ttypsin75. When the concentration of substrate is much larger than the amount of enzyme, the greatest impact on the reaction rate reaction is ________of the following reaction.A.ES−−−−→E+PB. E+S−−−−→E+PC. E+S−−−−→ESD. non above76. When the concentration of substrate is much larger than the amount of enzyme, the kinetic of the reaction is____.A. K catB. K mC. K cat/K mD. K m/K cat77. 10 µmol of enzyme catalytic 100 µmol of substrate in 5 minute，how many enzyme a catalytic constant?_____A.0.5µmolB.5µmolC.50µmolD.500µmol1. A2. C3. B4. D Multisubstrate reactions cotain :1. Sequential reactions which containsordered reactions and random reactions 2. ping-pong reactions.原版教材P1425. C 在乒乓反应中，第二个底物B与酶结合之前，第一个底物A已转变成产物P并被释放，在此过程中无三元复合体，反应物与产物“一进一出”。

Oxidation protection of CC–SiC composites by an

Journal of the European Ceramic Society 25(2005)577–587Oxidation protection of C/C–SiC composites by an electrophoretically deposited mullite precursorT.Damjanovi´c a ,∗,Chr.Argirusis a ,G.Borchardt a ,H.Leipner b ,R.Herbig b ,G.Tomandl b ,R.Weiss caTechnische Universität Clausthal,Institut für Metallurgie,Robert-Koch-Str.42,38678Clausthal-Zellerfeld,Germany b TU Bergakademie Freiberg,Institut für Keramische Werkstoffe,Gustav-Zeuner-Str.3,09596Freiberg,Germanyc Schunk Kohlenstofftechnik GmbH,Rodheimer Str.59,35452Heuchelheim,GermanyReceived 10October 2003;received in revised form 22March 2004;accepted 3April 2004Available online 20July 2004AbstractElectrophoretic deposition (EPD)is a suitable technique to produce mullite layers for acceptable oxidation protection of C/C–SiC bining sol–gel synthesis of 3Al 2O 3·2SiO 2mullite through hydrolysis and condensation of tetraethoxysilane (TEOS)and aluminum-tri-sec -butylate (Al-(OBu)3)with EPD yields sufﬁciently thick and homogeneous layers,which transform into mullite at 1300◦C.The protectiveness of the deposited mullite layers was tested in air in the temperature range 1300◦C ≤T ≤1550◦C by means of isothermal thermogravimetric analysis for up to 200h.The experimental data can be described by a phenomenological model of the (reduced)oxidation rate of the SiC layer underneath the outer mullite layer,which suggests that transport of carbon monoxide through mullite and silica is rate paring the oxidation rate of electrophoretically coated C/C–SiC samples to that of uncoated reference samples clearly demonstrates that mullite offers a signiﬁcant improvement to the oxidation resistance of the reference material.©2004Elsevier Ltd.All rights reserved.Keywords:Mullite;C/C–SiC;Composites;Electrophoretic deposition;Sol–gel processes1.IntroductionThe starting industrial material (Schunk Kohlenstofftech-nik GmbH,Heuchelheim,Germany)is a SiC coated carbon-reinforced carbon composite (C/C)in the form of thin slabs with average dimensions 20mm ×20mm ×2mm.The SiC layer is deposited by chemical vapour deposition (CVD)after the C/C substrate has ﬁrst been capillary inﬁl-trated with liquid silicon.As the CVD process takes place at high temperatures cracks develop in the SiC layer after cooling down to room temperature,which close again after renewed heating above 1100◦C.For short-term oxidation the resulting protective SiO 2layer is a suitable diffusion barrier but it degrades during prolonged service beyond 1300◦C.Furthermore,silica is not sufﬁciently resistant against erosion and chemical reactions with other compo-nents in the system (e.g.,ﬂue ash in power plants or steels in hardening cases for tool production).It is,therefore,∗Corresponding author.E-mail address:tanja.damjanovic@tu-clausthal.de (T.Damjanovi´c ).necessary to coat the SiC layer with more protective oxidesthan silica.Mullite ceramics are promising candidate materials for high temperature applications and oxidation protection of C/C–SiC composites.1–4Mullite satisﬁes all essential re-quirements for a refractory oxide layer on top of a C/C–SiC composite and has the corresponding mechanical proper-ties:similar low thermal expansion coefﬁcient (average val-ues for CTE of SiC and mullite in the temperature range 20◦C <T <1000◦C are:SiC:6.1×10−6K −1,mul-lite:4.4×10−6K −1),low thermal conductivity,excellent creep resistance,good chemical stability and low oxygen diffusivity.5,6Mullite coated SiC exhibits excellent oxida-tion resistance in dry air by forming a slowly growing native SiO 2scale under the mullite layer.Depending on the SiO 2content and on temperature mullite forms a low viscosity grain boundary ﬁlm which can close cracks and pores,thus offering a certain self-healing capacity.Plasma spraying or dip coating processes are mostly used to produce mullite layers from mixed oxide or mullite suspensions or from sol–gel systems,7,8respectively.The0955-2219/$–see front matter ©2004Elsevier Ltd.All rights reserved.doi:10.1016/j.jeurceramsoc.2004.04.005578T.Damjanovi´c et al./Journal of the European Ceramic Society25(2005)577–587sol–gel synthesis,however,represents a method to prepareultrapure and nano-sized mullite.Bulk ceramics obtainedfrom powders prepared this way reach nearly theoreticaldensity and have mullitization temperatures below1500◦C.9This paper demonstrates the possibility to prepare amullite precursor sol via chemical processing from whichelectrophoretic deposition(EPD)on C/C–SiC compositesproduces dense mullite coatings after sintering at1300◦C,which considerably improve the protectiveness of the pri-mary,SiC coating.2.Synthesis of the mullite precursor solProgress in the synthesis of chemically homogeneousmulticomponent oxides is particularly indebted to sol–gelscience.10,11However,it is well-known that hydrolysis andpolycondensation of silicon alkoxides are very slow com-pared to those of aluminium,and it is difﬁcult to achieve ahomogeneous co-polymerization.This may be reached byvery slow hydrolysis12,13of both alkoxides,by prehydroly-sis of the silicon alkoxide,14or by modifying the aluminiumalkoxide by a chelating group to reduce its reactivity.15,16Reagent grade chemicals used were alkoxides of Si andAl,tetraethoxysilane(TEOS;C8H20O4Si,VWR Interna-tional,p.a.)and Al(OBu)3(C12H27AlO3,VWR Interna-tional,p.a.),respectively,and isopropyl alcohol(C3H7OH,VWR International,p.a.)as a solvent.Due to the pronounceddifference in the reactivity of both alkoxides,acetylacetone(AcAc,VWR International,p.a.)was added as a chelatingagent inhibiting the condensation of Al(OBu)3.TEOS as aslower reacting precursor was prehydrolyzed with water ofdifferent pH values(pH2,6,10)adjusted by the additionof hydrochloric acid and ammonia.For the synthesis of suitable mullite precursors the con-centration of the separate alkoxide solutions,the pH of water(pH2,6,10),the molar ratio of the water to TEOS(r W= 2–10)and the concentration of acetylacetone were varied(0.01–1M)in order to obtain a series of different mulliteprecursor sols,whose electrokinetic properties were to beinvestigated systematically later on.The synthesis of Al(OBu)3–isopropanol solution was per-formed in a glove box so as to avoid uncontrollable at-mospheric moisture.Al(OBu)3was dissolved in isopropylalcohol on stirring,and acetylacetone was added into theAl(OBu)3solution until aﬁnal concentration0.1M wasreached.In the meantime,the separate solution of TEOSwas prepared.This reaction was performed in air becauseof the lower reactivity of the Si alkoxide.TEOS was prehy-drolyzed for1h by addition of water of deﬁned pH.The ad-dition of water was controlled by an automatic titrator(736GP Titrino,Metrohm,Germany).The interval of titrationwas10s and theﬂow rate was2ml min−1.Theﬁnal mullite precursor sol was obtained under con-trolled addition of the Al(OBu)3solution to the TEOSsol in a ratio corresponding to stoichiometric mullite (3Al2O3·2SiO2).Final mixing of both sols was performed in air with the Al(OBu)3sol covered with a paraﬁlm to avoid additional hydrolysis by atmospheric moisture.According to the mullite composition,a larger volume of Al(OBu)3 solution had to be mixed with a smaller volume of the TEOS solution.It would be technically easier to add the TEOS solution to the Al(OBu)3solution but this sequence leads to higher local concentrations of Al and induces in-homogeneities and phase separation in theﬁnal sol.Due to this fact,the Al(OBu)3solution was added in small volume increments to the TEOS solution.The addition was per-formed in volume increments of0.1ml,with aﬂow rate of 1ml min−1in intervals of10s,by an automatic titrator.The TEOS sol was stirred during the addition of the Al(OBu)3 solution,and the whole experiment was performed at room temperature.From the systematic investigations to optimize the above mentioned parameters of the sol–gel synthesis(concentra-tion of alkoxide solutions,pH value,amount of water needed for prehydrolysis of TEOS and amount of chelating agent), two promising experimental procedures were identiﬁed: (i)Synthesis of a mullite precursor sol was possibleunder the following conditions:concentration of alkoxide–isopropanol solutions is10vol.%,pH7(dis-tilled water),r W=2,addition of acetylacetone(c=0.1M).This precursor will be further denoted as MP1in the text.(ii)Synthesis of a mullite precursor sol with alkoxide–isopropanol solutions of10vol.%,pH2(adjusted by addition of HCl),r W=10and addition of acety-lacetone(c=0.1M).Due to the higher amount of water added for the hydrolysis of TEOS and prolonged co-polymerization of both alkoxides,we obtained a particulate sol suitable for the electrophoretic deposi-tion of the mullite precursor.This precursor will be further denoted as MP2in the text.3.Characterization of the mullite precursorsThe characterization of the stability and the mobility of the two mullite precursors prepared as described in Section2,was carried out by electrokinetic sonic anal-ysis(ESA-8000,MATEC Applied Sciences,USA).The average particle size of the synthesized mullite precur-sors MP1(d50≈20nm)and MP2(d50≤50nm),was estimated by means of transmission electron microscopy (TEM).The ESA measurements gave an ESA amplitude of7.80␮Pa m V−1(corresponding to a zeta-potential value ofζ=5.2mV)for the mullite precursor MP1and of 22.59␮Pa m V−1(ζ=15.1mV)for the mullite precursor MP2.Because of its low conductivity MP1was not suitable for the electrophoretic deposition,in contrast to MP2.In order to get information about the thermal behaviour of the precursors,as well as to determine structural changes asT.Damjanovi´c et al./Journal of the European Ceramic Society 25(2005)577–587579a function of the annealing regimes,XRD (Siemens D5000diffractometer),DTA/TG (Shimadzu Thermal Analyzer)and 29Si and 27Al MAS NMR (Bruker MSL 300MHz spectrom-eter)studies were performed.For the mullite precursor MP1the DTA/TG peak at 996.5◦C corresponded to mullite formation which was conﬁrmed by means of XRD analysis.This means that the obtained precursor sol was a single phase with both alkoxides mixed on a molecular scale leading to mullite crystallization at T ≤1000◦C.For the mullite precursor MP2,XRD investigations showed that the precursor remains amorphous up to approx-imately 1000◦C.The DTA/TG peak at 1129◦C corresponds to the formation of an alumina spinel also conﬁrmed by XRD.After a heat treatment at 1250◦C,according to XRD phase analysis,crystalline mullite is formed.According to the literature 16higher mullitization temperatures (T ≥1250◦C)indicate that the synthesized mullite precursor MP2is diphasic already in the sol state.These conclusions on the homogeneity of the synthesized mullite precursors were conﬁrmed by means of NMR analy-sis.The NMR results on the two different mullite precursors described above can explain the different mullitization tem-peratures.The detected chemical shifts and the respective signal assignments 17are given in Table 1.For a homogeneous mullite precursor a mullite-like Si surrounding,i.e.,Si(4Al)units should be present in the NMR spectra.The preliminary stage of the mullite pre-cursor MP1(heat-treated at 550◦C)and the ﬁnally formed sol–gel mullite from MP1at 1300◦C fulﬁl this expectation,whereas the non-crystalline phase from the mullite precur-sor MP2at 550◦C causes a broad signal in the spectrum.The peak maxima for both thermal stages of MP1(Fig.1a and c )are in the region for Si(4Al).The spectrum of the MP2precursor at 550◦C (Fig.1b )shows signiﬁcant evi-dence for silicon surrounded by less than four OAl groups,which triggers the formation of SiO 2rich and SiO 2deﬁ-cient domains in the annealed precursor.This ﬁnally results in higher mullitization temperatures than for the (more homogeneous)precursor MP1.Numerous 27Al NMR studies of aluminium-containing amorphous xerogels of oxide glasses have revealed the presence of three different aluminium environments.18Alu-minium nuclei in octahedral and tetrahedral environmentsTable 1Results of 29Siand27Al solid state NMR measurementsSample29SiMAS or CP/MASNMR 21chemical shift (ppm)Assignment of 29Si chemical shifts 27AlSATRAS NMR 22chemical shift (ppm)Assignment of27Alchemical shiftsMP2(550◦C)−75to –115(max.–99.2)Si(2Al) 3.4CN 6 CN 4(shoulder),CN 5present MP1(550◦C)−70to –112(max.–87.6)Si(4Al)26.1,2.0CN 5strongest signal CN 6 CN 4MP1(1300◦C)−86.1,shoulder –89,−93.3Si(3Al),Si(4Al)42.4,1.9CN 6∼CN 4,CN 4shoulder29SiNMR measuring conditions:MSL 300MHz,Larmor frequency 59.627MHz,CP time 5ms;27Al NMR measuring conditions:MSL 300MHz,Larmor frequency :co-ordination number of Al;CP/MAS NMR:cross polarization/magic angle spinning NMR spectroscopy;SATRAS NMR:satellite transition NMRspectroscopy.Fig.1.29Si solid state NMR:(a)CP/MAS spectrum of the preliminary stage of MP1(550◦C),(b)CP/MAS spectrum of MP2(550◦C)and (c)CP/MAS spectrum of MP1(1300◦C).give rise to characteristic chemical shifts of 0and 60ppm,respectively.The third species at 30ppm has been attributed to an [AlO 5]environment because its chemical shift lies between those of [AlO 6]and [AlO 4]and is very similar to the signal observed in well-known pentacoordinated Al compounds.Recently,it has been suggested by Schmücker and Schneider 19that in Al 2O 3-rich aluminium silicate gels and glasses with compositions close to mullite,this signal may arise from some tricluster [AlO 4]units.Taylor and Holland 20have related the aluminium coor-dination in aluminosilicate gels to their homogeneity,sug-gesting that Al(4)sites exist in regions of uniform Al/Si dis-persion and reﬂect the efﬁcient incorporation of aluminium into the tetrahedral silicate network.Less homogeneous regions containing discrete alumina-rich and silica-rich domains are believed to result in an increased proportion of Al(6)sites.By comparing the values of the composite quadrupolar coupling constant and of the isotropic chemical shift for the intermediate peaks in a mullite glass precursor and in crystalline mullite Bodart et al.18conﬁrmed that metastable aluminium atoms are present in the mullite glass precursor in an essentially pentacoordinated environment.Jaymes et al.21also suggested that the concentration of hexacoordinated Al atoms in single phase precursors prac-tically completely disappears before mullite crystallization at 980◦C.In diphasic precursors the aluminosilicate phase580T.Damjanovi´c et al./Journal of the European Ceramic Society25(2005)577–587has a higher content in AlO6octahedra,resulting in the crystallization of a spinel phase.The27Al NMR spectra22of the preliminary stage of the mullite precursor MP1,the mullite obtained from the precursor MP1after heating to1300◦C and the mullite precursor MP2show three Al signals assigned to different co-ordination numbers(CN)for each sample(Table1). From the results presented in Table1it is obvious that the mullite precursor MP2which is suitable for EPD has in its amorphous structure at550◦C a high concen-tration of hexacoordinated Al atoms.These Al(6)sites exist as a result of inhomogeneties at the Al–Si mixing scale and,according to this,MP2crystallizes into mullite at temperatures only above1250◦C.On the other hand, the mullite precursor MP1at550◦C has the strongest NMR signal for pentacoordinated aluminium which ac-cording to Schmücker and Schneider19acts as mullite nucleus because of a locally decreased activation energy. This precursor crystallizes directly into mullite below 1000◦C without formation of spinel,but is unfortunately unsuitable for electrophoretic deposition,as mentioned above.4.EPD of mullite from the precursor solAs the uniformity of the deposited layer is strongly de-pendent on the homogeneity of the surrounding electrical ﬁeld,23,24the following experimental set-up for EPD was chosen(Fig.2):•The substrate(C/C–SiC composite)was placed vertically in the mullite precursor sol with its surface parallel to the counter electrodes made of stainless steel.•Both the substrate and the counter electrodes were of pla-nar geometry.•The distance between the counter electrodes and the sub-strate was keptﬁxed at10mm.•The surface of the counter electrodes(40mm×60mm) was larger than the surface of the substrate(20mm×20mm),which yields a reasonably uniform electricalﬁeld near the substrate.•In order to avoid electrode marks on the substrate the position of the point contacts was changed before each individual deposition step.The topography of the sample surface was investigated with a surface proﬁler(Alpha-Step500,Tencor).The rough-ness of the surface25is characterized by the following param-eters:the average roughness,Ra=3.71␮m,and the max-imum peak-valley amplitude called total indicator run-out, TIR=50.02␮m.As explained above only the precursor MP2was suitable for our purposes and EPD was performed cathodically with different voltages and durations of deposition.The best per-formance of the deposited layer regarding the adhesion of the green layer to the C/C–SiC substrate after drying,wasFig.2.Experimental set-up for EPD of the mullite precursor. achieved with15V cm−1and15and30s deposition time. Under these conditions the deposited mass is directly pro-portional to the deposition time.The average mass gain per deposition cycle at15V cm−1and15or30s was0.16and 0.37mg cm−2,respectively.All C/C–SiC substrates were coatedﬁve-times with a sintering step between each depo-sition step at1300◦C(mullitization temperature obtained from XRD investigations on the mullite precursor MP2)in an atmosphere of99.996%Ar for2h.After theﬁnal sin-tering step the deposited layers were investigated by means of secondary electron microscopy(SEM)in order to obtain information about the thickness of the deposited layer and its composition.On the presented SEM micrograph(Fig.3)the deposited mullite layer is shown for the EPD conditions of15V/15s. According to the SEM micrograph,it is obvious that for the chosen EPD conditions the obtained layer adheres well to the surface of the substrate.The EDX analysis at the indicated spot showed a composition close to3/2-mullite.5.Thermogravimetric analysis of the oxidation rate in airThe protectiveness of the electrophoretically deposited mullite layers against isothermal oxidation in air in the tem-perature range from1300to1550◦C was investigated byT.Damjanovi´c et al./Journal of the European Ceramic Society25(2005)577–587581Fig.3.SEM micrograph of the cross-section of the electrophoretically deposited mullite layer on C/C–SiC at15V cm−1for15s(1×:EDX analysis spot).means of thermogravimetry(TG).The results are given in Table2.The experimentally obtained values for the oxidation rate of the mullite coated C/C–SiC samples were interpreted with the help of a phenomenological model whose basic version had been originally developed in our group by Fritze et al.1 for pulsed laser deposited mullite layers.According to this model(see Fig.4a),the samples exhibit a mass gain in the whole temperature range,due to the passive oxidation of the SiC layer.The formation of SiO2overcompensates the mass loss due to CO formation according to the reaction:SiC+32O2→SiO2+CO(1) The model will be applied to two situations,named case A and case B in the following.Table2Practical linear rate constants,k L,and practical parabolic rate constants, k P,for T≤1350◦C and for T≥1400◦C.The experimental values were, respectively,determined with the help of Eqs.(14)and(16)T(◦C)k L(mg cm−2h−1)k0L(mg cm−2h−1) H a(kJ mol−1) 1300 1.72×10−3 1.53×1083301350 3.74×10−3k P(mg2cm−4h−1)k 0P(mg cm−4h−1)1400 4.91×10−5 1.12×107362±66 1450 1.61×10−41500 1.90×10−41550 5.00×10−4Case A:The rate-limiting step is assumed to be trans-port of oxygen through the protective layer composed of the mullite layer and a native SiO2layer growing under-neath.Under the assumption that mullite does not substan-tially dissolve in the silica scale,the time(t)dependent thickness of the growing SiO2layer,x S(t),is obtained as follows:1With the atomic or ionic oxygenﬂux densities j M and j S in the mullite(M)and in the silica layer(S),respectively, j M=−˜D M c MRT∂µO∂x M(2a) j S=−˜D S c SRT∂µO∂x S(2b)the formal molecularﬂux density,j O2,of oxygen can be derived as follows:j O2=12j M=12j S=14˜D S c S˜D M c M˜D S c S x M+˜D M c M x S| µO2|RT(3)where˜D S,˜D M is the effective chemical diffusivity of oxy-gen O,(O2−or O2),in vitreous(or polycrystalline)SiO2and in polycrystalline mullite;x S(t),x M is the thickness of the silica layer and of the mullite layer;c S,c M is the molar con-centration of oxygen in silica and in crystalline3/2-mullite;µO is the chemical potential of oxygen(=1/2µO2); µO2 is the difference of the chemical potential of oxygen(seeFig.4a:µO2(I)−µO2(III)).582T.Damjanovi´c et al./Journal of the European Ceramic Society 25(2005)577–587Fig.4.(a and b)Schematic representation of the variation of the chemicalpotential of O 2and CO in the layer system mullite-SiO 2according to the case A (oxygen diffusion rate determining)and case B (CO diffusion rate determining),respectively.According to Eq.(1)the growth rate of the silica layer at the interface III is given by the following expression:d x S (t)d t =23V m ,SiO 2j O 2(4)which,together with Eq.(3),yields after integration:x S (t)=α(1+βt)−1 (5)where the parameters αand βare deﬁned as follows:α=213V m ,M V m ,S ˜DS ˜D M xM β=1696 V m ,S V m ,M 2| µO 2|RT 1x 2M ˜D 2M˜DS (7)where V m ,S ,V m ,M is the molar volume of silica and of mul-lite,respectively.As long as the growing SiO 2layer is much thinner than the deposited mullite layer (short oxidation times,i.e.,βt 1),oxygen diffusion through the mullite layer is the rate limitingstep and the oxidation kinetics is linear.Eq.(5)reduces to:x S (t)=12αβt =136V m ,S V m ,M | µO 2|RT ˜DM x Mt =k L t (8)withk L =136V m ,S V m ,M | µO 2|RT ˜DM x M(9)as the linear rate constant,which is directly proportional tothe oxygen diffusion coefﬁcient in the mullite layer,˜DM ,and inversely proportional to the thickness of the mullite layer,x M .Longer oxidation times (βt 1)lead to thicker SiO 2layers,and transition from the linear to the parabolic growth law will occur:x S (t)=αβ1/2t 1/2= 23| µO 2|RT˜D S 1/2t 1/2= 2k P √t(10)The parabolic rate constant k P is deﬁned as:k P =12α2β=13| µO 2|RT˜D S (11)and is directly proportional to the oxygen diffusion coefﬁ-cient in the SiO 2layer,˜DS .In agreement with the exper-iments this simple model shows that after long times the oxidation kinetics become parabolic.For the evaluation of the TG data,the oxidation rate can be expressed in terms of the mass change rate:1A 0d ( m sample )d t =ρS 3d x S d t =ρS 6αβ√1+βt (12)where A 0is the total (geometrical)surface of the sample,ρS the density of amorphous silica or ␤-cristobalite,respec-tively.For short oxidation times (βt 1)Eq.(12)yields a constant oxidation rate:1A 0d ( m sample )d t =ρS 6αβ=kL(13)which,after integration,yields a linear time dependency for the mass change: m sample A 0=kL t (14)with k L as a practical linear rate constant (for TG experi-ments).For longer oxidation times (βt 1)we obtain from Eq.(12)a time dependent oxidation rate:1A 0d ( m sample )d t =ρS 6α√β√t= k P 2t (15)which,after integration,yields a parabolic time dependency for the mass change: m sample A 02=2kP t (16)T.Damjanovi´c et al./Journal of the European Ceramic Society 25(2005)577–587583Table 3Calculated rate constants T (◦C)k L(mg cm −2h −1)k 0L(mg cm −2h −1) H a (kJ mol −1)1300 1.85×10−1 1.38×101135713504.30×10−1k Pfrom Eq.(18)with ˜D S from Eq.(19)after Rodriguez-Viejo et al.27:vitreous SiO 2k P(mg 2cm −4h −1)k 0P(mg 2cm −4h −1)1400 1.05×10−27.37×1073151450 2.03×10−21500 3.78×10−21550 6.78×10−2k Pfrom Eq.(18)with ˜D S from Eq.(20)after Rodriguez-Viejo et al.27:␤-cristobalitek P(mg 2cm −4h −1)k 0P(mg 2cm −4h −1)1400 2.80×10−3 4.0×10104211450 6.74×10−31500 1.54×10−21550 3.38×10−2Case A:Oxygen diffusion rate determining;k Lfrom Eq.(17)with ˜D Meff from Eq.(24)after Fielitz et al.5,6,26with k Pas a practical parabolic rate constant (for TG exper-iments).With the working Eqs.(17)and (18):kL =1318M S V m ,M | µO 2|RT ˜D M x M =ρS 3k L (17)k P=ρ2S 913| µO 2|RT ˜D S =ρ2S 9k P(18)oxidation rate constants can be calculated from diffu-sivity data for the linear and for the parabolic growth regime,respectively,and can be compared with experi-mentally obtained values (see Tables 2and 3)depending on the given oxidation time and temperature.For the mo-lar volume of mullite and for the densities of the vitreous silica and of ␤-cristobalite,respectively,the following values were adopted:V m ,M =134.81cm 3mol −1,ρam =2.2g cm −3,ρcr =2.27g cm −3,M S =60.082g mol −1is the molar mass of SiO 2.The thickness of the EPD–mullite layer was x M =7.5␮m (for EPD at 15V/15s).For the diffusion data for oxygen in single crystalline 2/1-mullite 5and polycrystalline 3/2-mullite,6,26in amor-phous silica and in ␤-cristobalite,27respectively,the follow-ing data set was used:SiO 2:see Rodriguez-Viejo et al.27D am =1.1×10−6exp −333kJ mol−1RT(m 2s −1)(19)D cr =5.6×10−4exp −439kJ mol−1RT(m 2s −1)(20)Mullite :see Fielitz et al.5,6,26D V =3.71×10−5exp−(433±21)kJ mol−1RT(m 2s −1)(21)D GB =6.2×10−3exp−(363±25)kJ mol−1RT(m 2s −1)(22)As has been shown,5there is only a small difference in the bulk diffusivity,D V ,measured in 2/1-mullite and in 3/2-mullite.For our calculations,we used an effective tracer diffusiv-ity,D Meff ,for the oxygen transport in polycrystalline mul-lite:D Meff =D V 1+4δd D GBD V(23)where δ≈1nm is the grain boundary thickness and d ≈35nm is the grain size,which was determined from XRD peak broadening.Inserting the values for δand d in Eq.(23)together with the data for volume and grain boundary tracer diffusion of oxygen in polycrystalline mullite,we get the following ex-pression for D Meff :D Meff =3.7×10−5e −(433±21)kJ mol−1/RT ×(1+19.1e (70±46)kJ mol−1/RT)(m 2s −1)(24)i.e.,D Meff (1573K)≈4×103D V (1573K).The calculated values of k L and k Pgiven in Table 3were obtained by setting ˜D S =D am (from Eq.(19))or ˜DS =D cr (from Eq.(20))and ˜DM =D Meff (from Eq.(24)).The value of µO 2is estimated as follows:the standard Gibbs energy of formation for the reaction in Eq.(1), G ◦1,can be expressed by the corresponding standard Gibbs en-ergies of formation for SiO 2,SiC and CO 28:G ◦1= G ◦SiO 2− G ◦SiC + G ◦CO(25)The equilibrium condition at the SiO 2/SiC interface (III)leads to:e − G ◦1/RT =a CO (III )a 3/2O 2(III )(26)with a CO ,a O 2being the activities of CO and O 2,respectively.In general,a CO (III)will be lower than unity.If,for the sake of simplicity,we set a CO (III )=1,we obtain:a O 2(III )=(e G ◦1(T)/RT )2/3(27)i.e.,µO 2RT =ln 0.2−23 G ◦1RT(28)for oxidation in air (a O 2(I )=0.2).In the temperature range under study,1573K ≤T ≤1823K, µO 2/RT varies between 39.84and 33.32.。

Mechanics of Vibrationmooc课后章节答案期末考试题库2023年

Mechanics of Vibration_东南大学中国大学mooc课后章节答案期末考试题库2023年1.When a two-degree-of-freedom system is subjected to a harmonic force, thesystemvibrates at the () ?参考答案:frequency of applied force2.The phenomenon of _____ can occur when the forcing frequency is close to thenatural frequency of the system.参考答案:resonance3. A two-degree-of-freedom system has () ?参考答案:two normal modes4.The equations of motion of a two-degree-of-freedom system are in general inthe form of () ?参考答案:coupled differential equations5.The magnification factor is also known as _____ factor.参考答案:amplification6.The equations of motion of a system will be _____ when principal coordinatesare used ?参考答案:uncoupled7.The number of degrees of freedom of a vibrating system depends on () ?参考答案:number of coordinates used to describe the position of each mass8.The theoretical basis of using energy method to find the natural frequency ofthe system is ().参考答案:Law of conservation of mechanical energy9.For the following free vibration system, which one decays the fastest ().参考答案:Critical damping system10.For discrete system, the governing equations are differential equations.参考答案:ordinary11.In proportional damping, the damping matrix is assumed to be a linearcombination ofthe _____ and _____ matrices.参考答案:mass, stiffness12.The _____ analysis is based on the expansion theorem.参考答案:modal13.The stiffness and flexibility matrices are related as ()?参考答案:[k] = [a]-114.Both boundary and conditions are to be specified to find the solution of avibrating continuous system.参考答案:initial15.The maximum acceleration of the simplified harmonic vibration of a springmass system is a, and the natural frequency of the system is ω. What is the maximum velocity of the system ? ()参考答案:a/ω16.An impulse force has a large magnitude and acts for a very _____ period oftime.参考答案:short17.The number of natural frequencies of a continuous system is参考答案:infinite18.The response of an undamped system under resonance will be() ?参考答案:infinity19.If a system is subjected to a suddenly applied nonperiodic force, the responsewill be_______?参考答案:transient20.Any nonperiodic function can be represented by a _____ integral.参考答案:convolution21.The complete solution of a vibration problem is composed of the _____ stateand transient solutions.参考答案:steady22.The amplitude of an undamped system varies with time. ()参考答案:正确23.The free-vibration equation of a string is also called a equation参考答案:wave24.Stiffness increases when the natural frequency (); The mass increases whenthe natural frequency ().参考答案:decrease increase25.The frequency of beating is () ?参考答案:ωn-ω26.The complex frequency response, H(iω),is defined by () ?参考答案:kX/F027.The fundamental natural frequency of a system is ()?参考答案:the smallest value。

2022年考研考博-考博英语-南京大学考试全真模拟易错、难点剖析B卷(带答案)第73期

2022年考研考博-考博英语-南京大学考试全真模拟易错、难点剖析B卷（带答案）一.综合题(共15题)1.单选题A visitor to a museum today would notice （）changes in the way museums are operated.问题1选项A.cognitiveB.rigorousC.conspicuousD.exclusive【答案】C【解析】考查形容词词义辨析。

cognitive “认知的”；rigorous “严格的，严厉的”；conspicuous “显著的”；exclusive “唯一的，独有的”。

句意：如果今天去博物馆参观，就会注意到博物馆的经营方式发生了显著的变化。

选项C符合题意。

2.单选题That man claimed to be a(n) （）of Confucius.问题1选项A.descendingB.ascendingC.descendantD.offspring 【答案】C【解析】近义词辨析。

Descending “下降”，ascending“上升”，descendant “后裔；子孙”，offspring “后代；产物”，offspring 作复数，句意：那个人自称是孔子的后代。

所以C符合题意。

3.单选题The goal is to use crops, weeds and even animal waste（）the petroleum that fuels much of American manufacturing.问题1选项A.in terms ofB.in favor ofC.in spite ofD.in place of【答案】D【解析】固定短语搭配。

in terms of “按照”；in favor of “有利于”；in spite of “尽管”；in place of“代替”。

句意：目标是利用农作物、杂草，甚至动物的排泄物代替石油来为美国制造业提供燃料。

真空无奇点黑洞的拟正则模：无质量标量场扰动

真空无奇点黑洞的拟正则模：无质量标量场扰动作者：奉勇辉刘道军来源：《上海师范大学学报·自然科学版》2015年第02期摘要：研究带有de Sitter中心的球对称黑洞在标量场扰动下的拟正则模.几何上来讲，这种黑洞在半径r很大的时候趋近与史瓦西解形式，在r 趋于0 的时候是de Sitter 解形式的.在这里通过变化参数r0 （它与宇宙学常数有关），角量子数l，泛音数n和黑洞质量M 来研究这个黑洞的拟正则模ω .计算结果发现：拟正则模在随r0 的变化出现极大值和极小值，同时对于一定范围的r0，当拟正则模随泛音数n变化时也会出现极值现象，这是值得关注的.关键词：真空无奇点黑洞；拟正则模； 6阶WKB近似中图分类号： P 142.9 文献标志码： A 文章编号： 10005137（2015）02012205黑洞是爱因斯坦广义相对论的重要预言之一.根据霍金和彭罗斯的黑洞奇点理论[1]，黑洞中不可避免地存在时空奇点.然而，奇点的存在使一切物理定律在这里都失效.因此，怎样避免黑洞中的奇点是非常重要的研究方向.1968 年Bardeen 构造了第一个无奇点黑洞，即存在事件视界的无奇点的黑洞，而且满足弱能量条件[2].尽管Bardeen 黑洞在理论上是自洽的，但它在物理解释上一直不令人满意.原因是爱因斯坦方程在中心处不存在真空解，如果要得到真空解必须引入额外的物质形式或者修改引力.直到AyonBeato 和Garcia[3]把它解释为与某种非线性磁单极子耦合的引力场，Bardeen 黑洞才得到大家的重视.1992年，Dymnikova根据一个特殊形式的球对称的真空能动张量获得爱因斯坦方程的一个精确解析解，当r 足够大时，它与史瓦西解一致，当r 很小时，它的性质与de Sitter 解类似，但它在任何地方都不存在奇点[4].这个爱因斯坦方程解没有用到电动力学或其他的理论，所以通过对这个黑洞的扰动的拟正则模的研究，将有助于了解对不带电的无奇点黑洞的性质.众所周知，存在于人们周围的每个物体当它们受到扰动时都产生专属于自己的振动的特征模.当黑洞受到物质场的扰动时，它会以引力波的形式产生辐射，从而产生属于自己的振动特征模.一般情况下只考虑线性微扰，这样扰动的能动张量在黑洞的背景度规中通过最低阶的线性近似才能被忽略不计.引力波在时空中的演化可以恰当地分为3个阶段，引力波的初始爆发阶段，此阶段依赖于扰动的初始条件，持续时间短，因此一般不作为研究对象.拟正则模阶段，属于较长时间的固有振荡的衰减期，它与初始条件无关，是黑洞的内禀振荡.晚期拖尾阶段，这个阶段引力波的衰减受到黑洞远处时空的反向散射，对于渐进平坦黑洞，拟正则模的衰减被幂律形式的衰减代替.近年来，黑洞（包括非广义相对论中的黑洞）的拟正则模得到了广泛研究比如[5-6]，这方面的详细综述可见[7-8].其原因首先源于实验的兴趣.在不久的将来引力波很可能被LIGO，VIRGO和LISA 等引力探测器探测到.由于拟正则模只与黑洞的物理性质有关，所以这些探测将有助于鉴别黑洞的物理特性.另一方面则由于Ads/CFT 对应.AdS/CFT为应对一些困难的问题提供了新的思考问题的角度，例如黑洞信息不守恒，黑洞奇点的性质和量子引力等[9].在计算拟正则模方面，主要有连分数法，WKB 近似，PoshlTeller 势近似法，半解析的单值法等等，这方面的详细综述可见[10].这里本文作者应用Konoplya 给出的六阶WKB近似公式计算拟正则模[11].3 结论鉴于黑洞在基础物理中的重要性，研究有黑洞的“声音”之称的拟正则模是很有意义的.本文作者研究了真空无奇点黑洞的标量场扰动并用6阶WKB方法计算了拟正则模在各种参数的变化下的数值，同时把结果跟史瓦西黑洞和Bardeen黑洞的拟正则模的相关结果做了比较.首先，质量M的增大，拟正则模的实部和虚部也同时减小.它们的变换与史瓦西黑洞的情况相同，同时Bardeen 黑洞的拟正则模也是相同的变换趋势.对于n=1，拟正则模的实部随角量子数l的增加呈线性增加，虚部随角量子数l的增加而减小，最终随角量子数l的增大趋于一个稳定值.Bardeen 黑洞的拟正则模也是相同的变换趋势.从图7和图8可以看出，当2r2o=1时和史瓦西黑洞（M=1）的拟正则模的虚部和实部近似重合，它们的变化可近似看做是线性的.结合图7～10，可以看到真空无奇点黑洞的拟正则模的实部和虚部大概在0参考文献：[1] HAWKING S W，PENROSE R.The singularities of gravitational collapse and cosmology[M].London：Proc Roy Soc Lon A，1970.[2] BARDEEN J M.Proceedings of GR5[M].USSR：Tbilisi，1968.[3] AYNBEATO E，GARCA A.The Bardeen Model as a Nonlinear MagneticMonopole[J].Physics Letters B，2000，493（1-2）：149152.[4] DYMNIKOVA I.Vacuum Nonsingular Black Hole[J].Gen Relativ Gravit，1992，24：235-242.[5] LI X Z，HAO J G，LIU D J.Quasinormal modes of string black hole[J].Phy Lett B，2001，507：312-361.[6] LIU D J，YANG B，ZHAI Y J，et al.Quasinormal modes for asymptotic safe blackholes[J].Class Quantum Gravity，2012，29：145009.[7] NOLLERT H P.Quasinormal modes：the characteristic ′sound′ of black holes and neutron stars[J].Class Quantum Gravity，1999，16：159-216.[8] BERTI E，CARDOSO V，STARINETS A O.Quasinormal modes of black holes and black branes[J].Classical Quantum Gravity，2009，26：163001.[9] HOD S.Bohr′s Correspondence Principle and the Area Spectrum of Quantum Black Holes [J].Phys Rev Lett，1998，81：4293-4296.[10] KONOPLYA R A，ZHIDENKO A.Quasinormal modes of black holes：From astrophysics to string theory[J].Rev Mod Phys，2011，83（3）：793-836.[11] KONOPLYA L R A.Quasinormal behavior of the Ddimensional Schwarzschild black hole and the higher order WKB approach[J].Phys Rev D，2003，68：024018.[12] CHANDRASEKHAR S.The Mathematical Theory of Black Holes[M].London：Oxford University Press，1992.[13] IYER S，WILL C M.Blackhole normal modes：A WKB approach.I.Foundations and application of a higherorder WKB analysis of potentialbarrier scattering[J].Phys Rev D，1987，35（12）：3621-3631.[14] FERNANDO S，CORREA J.Quasinormal modes of the Bardeen black hole：Scalar perturbations[J].Phys Rev D，2012，86：064039.Abstract：：In this paper，we investigate quasinormal mode（QNM） of the spherically symmetric BH with de Sitter centre due to scalar perturbations.This BH is，geometrically，asymptotically Schwarzschild for large r and asymptotically de Sitter as r → 0.We analyze the QNM ω by varying the parameter r0 （it is related to cosmological constant），spherical harmonic index l，overtune n and the BH mass M.According to calculations，we find that the QNM Lhave maximum and minimum with a changing r0，and for some r0 it is worthy to attention that the QNM have extreme with a changing overtune nKey words：vacuum nonsingular BH； QNM； 6order WKB approximation（责任编辑：顾浩然）。

the evaluation substrate will dictate the…

the evaluation substrate will dictatethe…The evaluation substrate will dictate the success and effectiveness of a study or experiment. It sets the foundation for how data is collected, analyzed, and interpreted. The choice of evaluation substrate can significantly impact the reliability and validity of research findings.评估基质将决定研究或实验的成功和有效性。

它为数据的收集、分析和解释奠定了基础。

评估基质的选择可以显著影响研究结果的可靠性和有效性。

The evaluation substrate refers to the specific tools, methodologies, or materials used to conduct an assessment or investigation. It encompasses various elements such as experimental design, measurement instruments, data collection techniques, and analysis approaches.评估基质是指用于进行评估或调查的具体工具、方法或材料。

它包括实验设计、测量仪器、数据收集技术和分析方法等各个要素。

Choosing an appropriate evaluation substrate requires careful consideration of several factors. Firstly, the nature of the research question or objective plays acrucial role in determining which evaluation substrate is most suitable. Different types of studies may require different substrates to effectively address their unique research goals.选择合适的评估基质需要仔细考虑几个因素。

安庆2024年03版小学5年级上册第2次英语第4单元暑期作业(含答案)

安庆2024年03版小学5年级上册英语第4单元暑期作业(含答案)考试时间：90分钟（总分:110）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题共100分)1. 填空题：The ______ (青蛙) has a long tongue to catch insects.2. 填空题：A snail can sleep for ______ (几个月).3. 听力题：The Earth's surface is shaped by both ______ and erosion.4. 选择题：What is the main ingredient in oatmeal?A. WheatB. RiceC. OatsD. Barley答案:C5. 听力题：The process of hydrolysis can affect the stability of ______.6. 填空题：A _____ (30) is an area with high elevation.7. 填空题：The capital of Jamaica is ________ (金斯顿).8. 填空题：The ________ (果蔬) are healthy snacks.9. 听力题：A _______ is an atom with an unequal number of protons and electrons.The butterfly has _______ (五彩斑斓) wings.11. 听力题：The cake has ______ (cherries) on top.12. 听力题：The _______ plays a vital role in the environment.13. 填空题：My pet fish swims in ______ (圆形) patterns.14. 选择题：Which season comes after winter?A. SpringB. SummerC. FallD. Rainy15. 选择题：What color do you get when you mix red and white?A. PinkB. PurpleC. OrangeD. Brown16. 选择题：What do we call the color of the sun?A. BlueB. YellowC. GreenD. Red答案: B. Yellow17. 填空题：A _____ (大树) can provide homes for many animals.18. 选择题：What is the main ingredient in popcorn?A. RiceB. CornC. WheatD. Soy19. 填空题：The ancient Romans had a complex system of ________ (道路).I enjoy _______ (参加)科学实验.21. 听力题：The chemical formula for lithium carbonate is __________.22. 听力题：His favorite movie is a ________.23. 选择题：What is the main ingredient in mayonnaise?A. OilB. EggsC. VinegarD. Mustard答案:A. Oil24. 选择题：What is the name of the famous waterfall located on the border between the U.S. and Canada?A. Angel FallsB. Victoria FallsC. Niagara FallsD. Iguazu Falls答案: C25. 填空题：The _____ (拼图) has many pieces to fit together.26. 听力题：The Earth's crust is constantly being reshaped by ______ forces.27. 填空题：The _____ (花粉) helps in the reproduction of plants.28. 听力题：A _______ is a substance that can increase the concentration of hydroxide ions in a solution.29. 选择题：What is the name of the famous monument in India?A. Taj MahalB. Qutub MinarC. Gateway of IndiaD. Red Fort答案:AWhat is the name of the famous ancient city in Greece?A. AthensB. SpartaC. DelphiD. Corinth答案: A31. 选择题：In which direction does the sun rise?a. Northb. Southc. Eastd. West答案:c32. 选择题：What do we use to write on paper?A. PaintB. PencilC. BrushD. Marker33. 填空题：I like to read about _______ (我喜欢读关于_______的书).34. 听力题：The process of ______ can lead to changes in landforms.35. 听力题：Salt is formed when an acid reacts with a _____.36. 听力题：A __________ is a mixture that can be separated by evaporation.37. 听力题：We like to _____ (explore/visit) new places.38. 听力题：A ____ is a small mammal that often hides in burrows.39. 填空题：My cat has sharp ______ (爪子).40. 选择题：Which bird is known for its colorful feathers?A. CrowB. PigeonC. PeacockD. Sparrow41. 选择题：What do we call the act of taking care of someone?A. CaringB. NurturingC. Looking afterD. All of the Above答案：D42. 填空题：The fox is very _________. (狡猾)43. 填空题：A ________ (种子) can stay dormant for years.44. 听力题：His favorite hobby is ________.45. 填空题：The rabbit nibbles on a _______ (兔子啃_______).46. 听力题：A chemical bond can form between ______.47. 听力题：A _______ is a reaction that requires energy input.48. 听力题：The capital of Spain is ________.49. 听力题：I have ___ apples in my bag. (three)50. 选择题：What is the process of water changing to vapor called?A. EvaporationB. CondensationC. PrecipitationD. Freezing51. 填空题：The ancient Romans were skilled in ________ (工程).52. 选择题：How do you say "love" in French?A. AmorB. LiebeC. AmourD. Szeretet53. 听力题：A __________ can often be seen hopping in grassy areas.54. 听力题：The bee gathers nectar from _______.55. 填空题：The _____ (花语) can convey different meanings.56. 选择题：What is the name of the famous wizard in the Harry Potter series?A. GandalfB. DumbledoreC. MerlinD. Harry Potter答案:D57. 填空题：I like to watch ______ (动画片) that are funny and entertaining.58. 听力题：I love to _______ (listen) to podcasts.59. 填空题：The _____ (根部) store energy for the plant.60. 选择题：What color is the sky on a clear day?A. BlueB. GreenC. YellowD. Red61. 选择题：What is the name of the famous inventor of the light bulb?A. Nikola TeslaB. Thomas EdisonC. Alexander Graham BellD. Henry Ford答案：B62. 填空题：The __________ (历史的轨迹) outlines paths.63. 听力题：The chemical equation for respiration includes glucose and _____.64. 填空题：The _____ (生态友好) practices help protect the environment.65. 听力题：We will eat _____ (pasta/rice) for dinner.66. 填空题：The first person to climb Mount Everest was ______ (希拉里).67. 选择题：What is the largest continent?A. AfricaB. AsiaC. North AmericaD. South America68. 选择题：What is the main ingredient in a smoothie?A. IceB. FruitC. YogurtD. Milk69. 听力题：The pizza is very ___ (cheesy).70. 选择题：What is the name of the fairy tale character who had long hair?A. RapunzelB. CinderellaC. Snow WhiteD. Sleeping Beauty答案:A71. 填空题：The flower has beautiful _______ (花有美丽的_______).72. 听力题：The chemical formula for potassium acetate is _______.73. 听力题：The gas used in fireworks for explosions is ______.74. 填空题：A ______ (蝴蝶) starts as a caterpillar.75. 选择题：How many letters are there in the English alphabet?A. 24B. 25C. 26D. 27答案: C. 2676. 填空题：The _______ (Aboriginal peoples) have lived in Australia for thousands of years.77. biodiversity) in rainforests is very high. 填空题：The ____78. 填空题：We should _______ (保持) our environment clean.79. 选择题：Which season comes after spring?A. WinterB. SummerC. AutumnD. Fall80. 听力题：My sister is a good ________.81. 听力题：His favorite sport is ________.82. 听力题：My ______ enjoys playing with his friends.83. 选择题：What is the name of the famous American holiday celebrated on the third Thursday of November?A. ThanksgivingB. Veterans DayC. Labor DayD. Independence Day答案:A84. 填空题：The goldfish swims in circles in its ______ (鱼缸).85. 选择题：What is the color of an eggplant?A. GreenB. YellowC. PurpleD. Red答案:C86. 填空题：My favorite toy is a ____ because it makes me laugh. (玩具名称)87. 填空题：The process of _______ separates mixtures based on different boiling points. (蒸馏)88. 填空题：I like to color pictures of _____.89. 填空题：I have a ________ that keeps me company.90. 填空题：The rain makes the ground _______.91. 填空题：My favorite dessert to bake is ______.92. 填空题：I call my teachers “.”93. 填空题：The __________ were ancient structures built in the Americas. (金字塔)94. 听力题：The atomic number of an element is determined by the number of ______ it has.95. 选择题：What do we call the science of matter and its changes?A. BiologyB. ChemistryC. PhysicsD. Environmental Science答案:B96. 听力题：The ancient Egyptians used ________ to record their stories.97. 听力题：A subatomic particle with no charge is called a _____.98. 填空题：My dad loves to watch __________ (体育比赛) on TV.99. 填空题：________ (兰花) are known for their beauty.100. 填空题：We have a ______ (大) celebration for our achievements.。

黑洞的准正模式(quasinormal modes)

Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identiﬁes the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great diﬃculty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR wereﬁrst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs wereﬁrst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will ﬁnd this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the ﬁeld.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a ﬁt to the fundamental and ﬁrst overtone QNM.In the next section we attempt to give a mathematical deﬁnition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In theﬁfth section we discuss the excitation and observation of QNMs andﬁnally in the sixth section we will mention the more signiﬁcant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asﬁnite strings,mem-branes,or cavitiesﬁlled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedﬁeld.) There is generally an inﬁnite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isﬁxed at its ends.All such systems can be described by systems of partial diﬀerential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of theﬁniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for diﬀerential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the diﬀerential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The deﬁnition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling theﬁnite string has solutions of various degrees of diﬀerentiability. To describe all“realistic situations”,clearly C∞functions should be suﬃcient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to inﬁnity.The simplest case are waves on an inﬁnite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice diﬀerentiable solution.No boundary condition is needed at inﬁnity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of theﬁelds are spatially compact, or–the other extreme–solutions with inﬁnite total energy for which theﬁelds grow at spatial inﬁnity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near inﬁnity! Again it turns out that mathematically it is more convenient to study all solu-tions ofﬁnite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that weﬁnd a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for inﬁnite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually diﬀerent to normal modes.To deﬁne quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisﬁes the ordinary diﬀerential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and deﬁneG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The deﬁnition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which deﬁnes a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be deﬁned as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical signiﬁcance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in ﬁnite parts of the space by a ﬁnite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions deﬁned by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On ﬁnite intervals and for late times the solution is approximated by a ﬁnite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near inﬁnity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Theﬁrst new point concerns the deﬁnitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,deﬁne the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at inﬁnity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an inﬁnite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largeﬁnite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for suﬃciently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniqueﬁngerprint which would lead to the direct identiﬁcation of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The diﬀerence between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisﬁes the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate deﬁned byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the eﬀective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testﬁelds as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingﬁeld.For“polar”perturbations the eﬀective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the deﬁnition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This deﬁnition is coming from the study of stellar pulsations in Newtonian theory and represents the type ofﬂuid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving diﬀerential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential diﬃculty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes diﬀerentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because inﬁnity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the deﬁning integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an inﬁnite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。

Linear_Control_Theory_ppt5

given any positive definite Hermitian (or any positive defined real asymmetric) matrix Q, there exists a positive definite Hermitian (or a positive definite real asymmetric) matrix P such that TP -P =-Q.
Chapter 2
Stability, controllability and observability
1
Stability
2
Controllability
3
Observability
1 Stability
For continuous-time system, the state space model is d
u (k 1) u (k 2) [ 2 k 1] 1 [ X (k ) k X (0)] u ( 0)
Stability
Liapunov stability analysis plays an important role in the stability analysis of control systems described by state space equations.
From the classical theory of mechanics, we know that a vibratory system is stable if its total energy is continually decreasing until an equilibrium state is reached.

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a r X i v :g r -q c /0605013v 1 1 M a y 2006Stability and quasinormal modes of the massive scalar ﬁeld around Kerr black holesR.A.Konoplya ∗and A.V.Zhidenko †Instituto de F´ısica,Universidade de S˜a o Paulo C.P.66318,05315-970,S˜a o Paulo-SP,BrazilWe ﬁnd quasinormal spectrum of the massive scalar ﬁeld in the background of the Kerr blackholes.We show that all found modes are damped under the quasinormal modes boundary conditions when µM is not large,thereby implying stability of the massive scalar ﬁeld.This complements the region of stability determined by the Beyer inequality for large masses of the ﬁeld.We show that,similar to the case of a non-rotating black holes,the massive term of the scalar ﬁeld does not contribute in the regime of high damping.Thereby,the high damping asymptotic should be the same as for the massless scalar ﬁeld.PACS numbers:04.30.Nk,04.50.+hI.INTRODUCTIONScalar ﬁeld in a black hole backgound is a subject of active investigation [1],yet the massive scalar ﬁeld has been investigated only in very few studies as to its quasi-normal spectrum [2],[3],late-time behavior [4]and scat-taring properties [5].Behavior of the massive scalar ﬁeld in a black hole background is quite diﬀerent from that of the massless one in many aspects:ﬁrst,it demon-strates the so-called superradiant instability [6],which is absent for massless ﬁelds.This super-radiant regime happens provided the real oscillation frequency satisﬁes the inequality |Reω|<ma/(2Mr +),where m is the az-imutal number of the mode,and M is the mass of the black hole.Another research,indicating the instability of massive scalars was that by Detweiler [7]where slowly growing instability was suggested for values µM <<1,i.e.in the regime,when particle Compton wavelength is much larger then the size of the black hole (µis the mass of the scalar ﬁeld).Second,it may cause existence of inﬁnitely long-living modes called quasi-resonances [8].Finally,at asymptotically late times the massive ﬁelds show universal behavior independent on spin of the ﬁeld [9].Note,that scalar ﬁeld with the mass term can be also interpreted as a self-interacting scalar ﬁeld within regime of small perturbations [10].One more recent ap-plication of dynamic of the massive scalar ﬁeld in a black hole backgound comes from diﬀerent brane-world theo-ries where massless ﬁelds can be considered as eﬀectively massive ﬁelds on the brane.Yet,the strongest motivation to make an extensive search for quasinormal modes of massive scalar ﬁeld is that until now the stability for massive scalar ﬁeld pertur-bations in the Kerr background is not studied under non-super-radiant (quasinormal)boundary u-ally,stability of spherically symmetric black holes,like Schwarzschild or Reissner-Nordrstrom ones,can be rela-tively easily proved by using positivity of the conserved2Mr +r ++a22 II.BASIC EQUATIONSThe background metric in the Boyer-Lindquist coordi-nates t,r,θ,φhas the form:ds2= 1−2MrΣdtdφ−ΣΣ sin2θdφ2,(2)where,M is the mass of the black hole,∆=r2+a2−2Mr,andΣ=r2+a2cos2θ.The massive scalarﬁeld obeys the equation:Φ+µ2Φ=0.(3)The radial wave equation(Φ=e−iωt e imφR(r)S(φ,θ))iswell-known:ddr +K2sinθddθ +−m22χr+,R(r→r+)∝(r−r+)−iα,α=ωr+(r2++a2)−mar+M2−a2,andχ=±r+−a2rr+−a2 −iα××∞i=0a i rr+−r2+ a n=γnβn−1−αn−2γn−1αn, a n+1βn+1−αn+1γn+2βn−1−αn−2γn−1βn+1−αn+1γn+2Table I:Values of the quasinormal frequencies for the fundamental mode n=0,ℓ=m=0for diﬀerent values of massµ,and rotation a.ℓ=m=0µ=0.1µ=0.2µ=0.3a Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)10.330086-0.267718i0.325510-0.288297i20.292386-0.489967i0.296315-0.501634i30.266009-0.720273i0.270472-0.726246i40.245565-0.952427i0.249182-0.955699i50.228436-1.184966i0.231248-1.186916i100.164599-2.343964i0.165873-2.344194i150.117999-3.499102i0.118599-3.499165iresonances.Therefore,when increasingµ,the strict linein Fig.3does not have continuation into negative half-plane.We see here that,the larger mass of theﬁeldµis,the more low laying modes will change into quasi-resonances.Meanwhile,the rest of the spectra consistsof ordinary modes(i.e.modes withﬁnite time of damp-ing).Moreover,at suﬃciently high overtone number,the diﬀerence between massive and massless case will beneglected(see Table II).This behavior mimics the QNbehavior of non-rotating black holes.Yet,potential formassive scalarﬁeld around non-rotating black holes is-6-4-2246m-0.0956-0.09555-0.0955-0.09545-0.0954-0.09535-0.0953Im wFigure 1:Imωas a function of azimutal number m for ℓ=6,a =0.3,n =0,µ=0.1.-6-4-2246m1.151.21.251.31.351.4Re w Figure 2:Imωas a function of azimutal number m for ℓ=6,a =0.3,n =0,µ=0.1.deﬁnitely positive everywhere outside black hole,while for Kerr solution,the potential depends upon complexfrequency and such a stightforward analysis is impossi-ble.Now we are in position to state that the high damping asymptotic for quasinormal modes of massive scalar ﬁeld is the same as for massless one,i.e.the real part increases with m according to the numerical law described in [20].To exclude possible instability we should search for QNMs for diﬀerent values of ℓand m .As one can see from representative plots Fig.1,at large ℓthe imagi-nary part of the QN frequencies remains bounded and,thereby,does not show any tendency to instability.The dependence of the real osicillation frequency on the az-imutal number m can be found on Fig.3.After careful investigating of region of small values µM ,we conlcude that there are no unstable modes .As to the case of intermediate µM ,our extensive search of quasinormal modes implies that there are no unstable modes at least as far as µM is not much larger then 1.We certainly cannot ﬁnd quasinormal modes for any large value of µ.This happens because the numerical procedure used here requires slow changing of µ,when searching for new modes,in order not to jump occassion-aly into another overtone.Thereby,considerably large values of µcan be reached only for very long time of computation.Table III:Values of the quasinormal frequencies for the fundamental mode n =0,ℓ=1,m =0for diﬀerent values of mass µ,and rotation a .ℓ=1;m =0µ=0.1µ=0.2µ=0.3aRe(ω0)-Im(ω0)Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)Table IV:Values of the quasinormal frequencies for the fundamental mode n =0,ℓ=1,m =1for diﬀerent values of mass µ,and rotation a .ℓ=1;m =1µ=0µ=0.1µ=0.2aRe(ω0)-Im(ω0)Re(ω0)-Im(ω0)Re(ω0)-Im(ω0)00.2646890.3028510.2286500.5385990.2025580.7876320.10.2742600.3017110.2391560.5346700.2128980.7808590.20.2848720.2999360.2508830.5294660.2245370.7720460.30.2967380.2973320.2640790.5226330.2377490.7606440.40.3101400.2936080.2790720.5136400.2528940.7458200.50.3255100.2882970.2963150.5016350.2704710.7262460.60.3434410.2806030.3164810.4851690.2912440.6996640.70.3649320.2690190.3406440.4615050.3165110.6618350.80.3917900.2501870.3707800.4245420.3489800.6035570.90.4281120.2138330.4117820.3558390.3821030.6444250.950.4416340.2871110.4316640.3991490.3821030.6444250.990.4736040.2756550.4681240.3951390.4607870.5761990.9950.4806130.2931360.4776640.3816870.4711220.6047830.080.0850.090.0950.10.1050.11Re w 0.050.10.150.20.250.3Im wΜ 0.1Μ 0.2Μ 0.3Μ 0.4Μ 0.5Μ 0.6Μ 0.7Μ 0.8Μ 0.9Μ 1.Μ 1.1Μ 1.2Μ 1.3Μ 1.4Figure 3:Reωvs Imωfor n =1,a =0.5,ℓ=m =0fordiﬀerent values of µquasinormal modes for values µM not much larger then 1.In this regime all found low laying modes are damped.We have shown that similar to the Schwrazschild case,the massive term does not contribute into the high damp-ing part of the spectrum.Therefore we can conclude that for not large values of µM there is no instability for mas-sive scalar ﬁeld under the quasinormal mode boundary conditions.From the inequality (1)it follows that for a given black hole mass M and rotation a ,for any large µ,there is suﬃciently large m ,so that inequality is not performed.Therefore we sould have instability for suﬃ-ciently large values of m ,even for large µ.Yet,we have whon in Fig.1,that QN spectrum for larger values of m does not show any tendency to instability,because the Imωfor increasing values of m remains bounded within some region of negative values corresponding to damp-ing.Thefore inequlity (1)found in [11]means stability for ﬁelds with suﬃciently large values of µ.In addition,we have shown that the qusi-resonances,which exist for the Schwrazschild metric,exist also for 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