jianmolunwen2
Commutative weakly distance-regular digraphs of girth 2
In [3], we defined weakly distance-regular digraphs and determined all commutative 2valent weakly distance-regular digraphs. In [2], Suzuki determined all thin weakly distanceregular digraphs and proved the nonexistence of noncommutative 2-valent weakly distanceregular digraphs. In [4], we gave a classification of commutative weakly distance-regular digraphs with girth 2 and k − k1,1 = 1. The proof of the main results in [4] strongly relies on the commutativity of weakly distance-regular digraphs. This paper address that the condition ‘commutativity’ may be removed. The following is our main result. Theorem 1.1 Let Γ be a weakly distance-regular digraph of valency k, diameter d, and girth 2. If k − k1,1 = 1, then Γ is isomorphic to one of the following digraphs: (1) Cay(Z4 , {1, 2}). (2) Kn [C4 ]. (3) ∆[C3 ]. (4) ∆ × Cr , r is odd or d < r/2. Here Kn is a complete graph with n vertices, Cr = Cay(Zr , {1}), and ∆ is a distanceregular graph. In particular, all weakly distance-regular digraphs with girth 2 and k − k1,1 = 1 are commutative.
lunwen2
´222T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229by measuring the absorption at one or more wavelength by a spectrophotometer[1,2]. Spetrophotometric methods determine exactly the amount of pigments but they are unsuitable to differentiate between the various pigment fractions[3].Various liquid chromatographic methods have been frequently employed for the separation and quantitative determination of pigments.Thus,thin-layer chromatography(TLC)was used for the separation of pigments of Capsicum annuum[4]and red wine[5].The color substances of many foodstuffs such as black tea liquors[6],turmeric[7],caramel[8], etc.were determined by high performance liquid chromatography(HPLC).Supercritical fluid chromatography[9]and centrifugal partition chromatography[10]have found only limited application in the analysis of pigments.As the pigment composition is characteristic for the food[11]and it depends on the environmental conditions[12]the pigment composition can be used for the identification of the origin and/or the treatment of the food.Principal component analysis(PCA)has been developed for the evaluation of large data matrices[13].PCA calculates the similarities and dissimilarities between the columns and rows of a data matrix of any dimensions without being one column or one row the dependent variable.Although PCA reduces the dimensionality of the original data matrix the resulting matrices of PC loadings and PC variables are sometimes even multidimensional making difficult the visual evaluation of the results.In order to facilitate the evaluation the dimensions of the matrices of PC loadings and variables can be reduced either to two by non-linear mapping technique[14]or to one by cluster analysis[15].Due to its simplicity PCA has been frequently used in food science and technology for the classification of foodstuffs according to their chemical composition. Thus,PCA has been successfully employed for the differentiation between Rias Baixas and non-Rias Baixas wines according to the composition of volatile aroma compounds determined by gas chromatography(GC)[16,17],for the assessment of the authenticity of edible oils according to the composition of fatty acids determined by GC[18],and for the multivariate identification of natural triglyceride oils[19].The objectives of this work were tofind the best separation procedure for the color pigments of chili powder,to determine the pigment composition of chili powders by this TLC method and to employ principal component analysis for the differentiation of chili powders according to the composition of pigments.Although the separation capacity of HPLC is considerably higher than that of TLC it needs complicated instrumentation and parallel samples cannot be simultaneously analysed.Because of its rapidity and simplicity TLC has been chosen for the separation of the pigments of chili powder.To the best of our knowledge the pigments of chili powders have not been studied in detail.2.Material and methodsPlastic sheets coated with neutral aluminium oxide60F(Type E);aluminium254sheets coated with silica gel60,kieselguhr F cellulose,and polyamide;and glass254plates coated with cyano F S,diol F S,and amino F S were obtained from Merck254254254(Darmstadt,Germany).All except the polyamide plates were used,without pretreatment, for adsorption TLC.For reversed-phase chromatography polyamide,cyano,diol and´T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229223 amino plates were used as received;the others were impregnated by overnight pre-development in n-hexane–paraffin oil95:5v/v.The separation capacity of the following mobile phases was tested in adsorption TLC:n-hexane:carbon tetrachloride(99:1,v/v), n-hexane–tetrahydrofuran(76:24,48:52,and24:76,v/v).Water:methanol:acetonitrile (85:12:3,65:28:7,45:44:11,and3:77.5:19.5,v/v/v),water:acetone(10:90,15:85, 50:50,and65:35,v/v),water:tetrahydrofuran(10:90,20:80,30:70,40:60,50:50,60:40, 65:35,70:30,75:25,and80:20,v/v),water:dioxane(10:90,25:75,50:50,and65:35, v/v),water:acetonitrile(15:85,50:50,and65:35,v/v),water:methanol(5:95,35:65,and 50:50,v/v),1M acetic acid:acetone(10:90,v/v),1M acetic acid:tetrahydrofuran (50:50,v/v),1M sodium acetate:tetrahydrofuran(50:50,v/v),1M sodium acetate:acetone(10:90,v/v),1M sodium chloride:tetrahydrofuran(50:50,v/v), water:acetone(15:85,v/v)containing10mg/mL of sodium dodecyl sulfate(SDS)were used in the reversed-phase separation mode.The use of SDS was motivated by the finding that SDS can modify the reversed-phase retention behavior of organic com-pounds[20].The pigment composition of six chili powders of different origin was determined(sample I:Thailand;sample II:India;sample III:China;sample IV: Malaysia;sample V:Bali;sample VI:Pakistan).Color pigments were extracted by shaking3g of chili powder with10ml of acetone for30min.The suspension was centrifuged and the supernatant decanted.This procedure was repeated as the solid rest was nearly white.The collected supernatants were evaporated to1mlfinal volume,and 3m l of this solution was applied to the TLC plates.Because of the good solubility of carotenoid pigments in acetone it has been frequently used for the extraction of color pigments of paprika powders[21,22].As paprika and chili contain similar color pigments it was assumed that acetone will be an adequate extracting agent for the pigments of chili powders too.Plates were developed in sandwich chambers,the developing distance being ca.15 cm.As the pigments are sensitive to light[23],developments were carried out in dark. After development the plates were inspected visually and then the plates showing good separation were evaluated by a Shimadzu CS-930Dual Wavelength TLC Scanner (Japan)at340and440nm to detect both yellow and red pigments near to their absorption maxima[24,25].The intra-day and inter-day reproducibilities of the R values and peak areas wereFdetermined by the analysis of the same sample four times a day duringfive consecutive days and the relative standard deviations(R.S.D.%)have been calculated.Principal component analysis was carried out by taking into consideration the quantity of the seven largest pigment fractions(absolute peak areas without normalisation).As the concentrations of other pigment fractions were low their inclusion in the calculations probably do not influence markedly the results.We are well aware that the absorption ratios of pigments are not necessarily identical with their concentration in the chili powder.However,it has been assumed that the chili powders can be classified even by using the absorption ratios of pigments without the exact knowledge of their con-centration.PCA was performed twice:on the data obtained on impregnated diatomace-ous earth plates in eluent acetone–water17:3v/v,and on the data obtained on impregnated diatomaceous earth plates with the eluent acetone–water17:3v/v containing10mg SDS/ml.The six chili powders were the variables and the quantity of´224T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229the seven pigment fractions determined at440nm(observations1–7)and at340nm (observations8–14)were the observation in both cases.The limit of the variance explained was set arbitrary to99%.The two dimensional non-linear map of the PC loadings and variables were also calculated.The iteration of the non-linear map was carried out to the point when the difference between the two last iterations was lower 8than10.Softwares for PCA and non-linear mapping was developed by Dr.Barna ´Bordas(Plant Protection Institute,Hungarian Academy of Sciences,Budapest,Hun-gary).The algorithm has been prepared according to the theoretical bases described in Ref[13].3.Results and discussionThe pigments of chili powder cannot be separated in most of the adsorption TLC systems,some fractions generally remained on the start.Interestingly,tetrahydrofuran proved to be the strongest eluent,it was able to move each pigment fraction on the silica layer.According to the traditional solvent strength system tetrahydrofuran is a weaker solvent than acetone and acetonitrile.However,in this special case it showed higher elution capacity than acetonitrile and acetone did.This discrepancy cannot be explained by the insolubility of pigments in any solvents.We have to admit that we do not have any valid theory to explain the anomalous elution behavior of tetrahydrofuran.The best separations using adsorption TLC was achieved on silica layers using eluents composed of hexane–tetrahydrofuran mixtures.As it has been established many times that TLC can be used as a pilot method for high-performance liquid chromatography(HPLC) [26,27],the adsorption HPLC separation of the pigments of chili powder can be obtained on silica column using n-hexane–tetrahydrofuran gradient elution.(Our recent experi-ments proved that pigments can be separated into more than30fractions on a silica column using n-hexane–tetrahydrofuran gradient).In most instances a large amount of the pigment material generally remained at the origin in reversed-phase TLC also.Thisfinding indicates that pigments contain a highly hydrophobic fraction which binds strongly to the apolar surface of the support.The retention of pigment did not depend on the acidic or alkaline character of the eluent suggesting that pigments did not contain dissociable substructures.The best reversed-phase separations were achieved by using acetone:water17:3v/v eluent with and without SDS on impregnated diatomaceous earth support.Seven main pigment fractions were found,the largest two fractions were near to the start and near to the front(Fig.1). The mobility of fractions2–6decreased in the presence of SDS but the separation efficiency of the TLC system remained the same,that is in this instance the use of SDS did not improve the separation(data not shown).It has to be mentioned that the use of SDS is not preferred when the fractions have to be extracted from the plates for identity purposes.The absorption ratio of the fractions was different at440and340nm indicating that the visible spectra of the individual pigments is different.Thisfinding made necessary the evaluation of the plates at two wavelengths.As it has been previously mentioned the pigments of chili powder have never been´T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229225Fig.1.Separation of the colour pigments of chili powder on impregnated diatomaceous earth eluted with acetone–water(17:3v/v).studied in detail,therefore,reliable information of the chemical structure of pigments is not available.As paprika and chili are closely related it can be assumed that the pigments of chili are also carotenoid derivatives but they are not necessarily identical with the pigments of paprika.The R.S.D.%values were between3.4and5.3%and6.1and8.6%for the R valuesFand peak areas,respectively.The R.S.D.%of R values proves again that theFreproducibility of TLC is lower than that of HPLC.The relatively high R.S.D.values of peak areas can be tentatively explained by the fact that the baseline separation of some peaks was not achieved which lessens the accuracy of the integration process.The results of PCA using the data obtained on impregnated diatomaceous earth with acetone–water17:3v/v as eluent are compiled in Table1.Two principal components explain the majority of variance indicating that the six original variables can be substituted by two background(abstract)variables with only1.54%loss of information. Unfortunately,PCA does not prove the existence of such background variables as concrete physicochemical entities,but only indicates their mathematical possibility.The fact that each chili powder has a high loading in thefirst PC indicates the similarity of pigment composition of chili powders.The situation is highly similar in the case of eluent containing SDS(Table2).The data prove again the similarity of chili powders and suggests that the addition of SDS to the eluent does not influence considerably the pigment composition.The two-dimensional non-linear maps of PC loadings are shown in Fig.2.Chili´226T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229Table1Similarities and dissimilarities between the pigment composition of chili powders,results of principal component analysis,eluent:acetone–water17:3v/vNo.of PC Eigenvalue Variance Total varianceexplained%explained% 1 5.0183.5283.5220.8113.5097.01 Principal component loadingsNo.of chili No.of principal componentpowder12I0.9720.22]II0.580.81]III0.9720.20]IV0.9920.08]V0.940.19]VI0.9620.17]powders except sample II form a distinct cluster on both maps.These results indicates that the pigment composition of chili powder II markedly deviates from those of the other chili powders,that is reversed-phase TLC combined with PCA is suitable for the classification of chili powders according to the pigment composition.Pigment fractions 2–6and9–13form a cluster on the two-dimensional non-linear map of PC variables (Fig.3).Thisfinding suggests that the determination of two main fractions near to the start and to the front at440and340nm characterizes the chili powders.It can be concluded from our data that the color pigments of chili powders can be separated by reversed-phase thin-layer chromatography.The ratio of pigments measuredTable2Similarities and dissimilarities between the pigment composition of chili powders,results of principal component analysis,eluent:acetone–water17:3v/v containing10mg SDS/mlNo.of PC Eigenvalue Variance Total varianceexplained%explained%1 5.1185.1285.1220.8013.3498.46Principal component loadingsNo.of chili No.of principal componentpowder12I0.9920.04]II0.530.85]III0.9820.19]IV0.9820.11]V0.9820.18]VI0.990.05]´T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229227Fig.2.Similarities and dissimilarities between the pigment composition of chili powders.Two-dimensional non-linear map of principal components loadings.Numbers refer to chili powders in Section2.A:eluent23acetone–water17:3v/v;number of iterations5162;maximum error52.01310.B:eluent acetone–water2517:3containing10mg SDS/ml;number of iterations547;maximum error54.18310.at440and340nm is characteristic for the individual chili powders.Principal component analysis is a useful tool to detect the similarities and dissimilarities between the pigment composition of chili powders and to classify the powders according to the pigment composition.This method makes possible the rapid and accurate classification of chili powders of various origin taking into consideration the quantity and distribution of pigments.The simple and rapid method can be used for future chemotaxonometric studies using thefingerprint of color pigments determined by reversed-phase thin-layer chromatography.´228T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229Fig.3.Similarities and dissimilarities between the pigment fractions of chili powders.Two-dimensional non-linear map of principal components variables.Numbers refer to pigment fractions in Section2.A:eluent24acetone–water17:3v/v;number of iterations5200;maximum error56.25310.B:eluent acetone–water2417:3containing10mg SDS/ml;number of iterations5280;maximum error55.97310.4.Simplified description of the method and its applicationReversed-phase thin-layer chromatography combined with principal component analysis and subsequent two-dimensional non-linear mapping proved to be a suitable method for the discrimination among chili powders according to the composition of color pigments.This method makes possible the rapid and accurate classification of chili powders of various origin taking into consideration the quantity and distribution of pigments.The simple and rapid method can be used for future chemotaxonometric studies using thefingerprint of color pigments determined by reversed-phase thin-layer chromatography.AcknowledgementsThis work was supported by the Portugues–Hungarian cooperation project‘Develop-ment of new methods and their application for the assessment of the effect of environmental conditions on the stability of color pigments in foods’.References[1]Conrad RS,Sundstrom FJ,Wilson PW.Evaluation of two methods of pepper fruit colour determination.Hortscience1997;22:608–9.´T.Cserhati et al./J.Biochem.Biophys.Methods45(2000)221–229229[2]Navarro F,Costa J.Note.Evaluation of paprika pepper colour by triestimulus colorimetry.Rev EspCienc Tecnol Aliment1993;33:427–34.`´´[3]Minguez-Mosquera MI,Jaren-Galan M,Garrido-Fernandez J.Color quality of paprika.J Agric FoodChem1992;40:2384–8.´´´[4]Cserhati T,Forgacs E,Hollo J.Separation of color pigments of Capsicum annuum by adsorption andreversed thin layer chromatography.J Planar Chromatogr-Mod TLC1993;6:472–5.[5]Shi R,Schwedt G.HPLC-und DC-Analytik phenolischer Inhaltstoffe im Wein.Deutsch LebensmRundsch1995;91:14–7.[6]Bailey RG,Nursten HA,McDowell parative study of the reversed-phase high-performance liquidchromatography of black tea liquors with special reference to the thearubigins.J Chromatogr 1991;542:115–21.[7]Taylor SJ,McDowell I.Determination of the Curcuminoid pigments in turmeric(Curcuma domesticaVal)by reversed-phase high-performance liquid chromatography.Chromatographia1992;34:73–7.[8]Coffey JS,Castle L.Analysis of caramel colour(Class III).Food Chem1994;51:413–6.[9]Gere DR.Separation of paprika oleoresin and associated carotenoids by supercriticalfluid chromatog-raphy.Hewlett Packard Application Note,1983;AN800-5.[10]Renault JH,Thepenier P,Zeches-Hanrot M,Foucault AP.Separation of the two major anthocyanins fromchampagne vintage by-products by gradient centrifugal partition chromatography.J Liq Chromatogr 1995;18:1663–70.´[11]Almela L,Lopez-Roca JM,Candela ME,Alcazar MD.Carotenoid composition of new cultivars of redpepper for paprika.J Agric Food Chem1991;39:1606–9.´[12]Almela L,Lopez-Roca JM,Candela ME,Alcazar MD.Separation and determination of individualcarotenoids in a Capsicum cultivar by normal-phase high-performance chromatography.J Chromatogr 1990;502:95–116.[13]Mardia KV,Kent JT,Bibby JM,editors,Multivariate analysis,London:Academic Press,1979,pp.213–54.[14]Sammon Jr.JW.A non-linear mapping for data structure analysis.IEEE Trans Comput1969;C18:401–7.[15]Dillon WR,editor,Multivariate analysis,New York:John Wiley and Sons,1984,pp.157–208.[16]Garcia-Jares C,Garcia-Martin S,Cela-Torrijos R.Analysis of some highly volatile compounds of wineby means of purge and cold trapping injector capillary gas chromatography.Application to the differentiation of Rias Baixas Spanish white wines.J Agric Food Chem1995;43:764–8.[17]Garcia-Jares C,Garcia-Martin S,Carro-Marino N,Cela-Torrijos R.GC–MS identification of volatilecomponents of Galician(Northwest Spain)white wines.Application to differentiate Rias Baixas Wines from wines produced in nearby geographical regions.J Sci Food Agric1995;69:175–84.¨¨¨[18]Schwaiger I,Vojir F.Anwendung multivariater statistischer Verfahren zur Uberprufung der Authentizitat¨von Speiseolen.Deutsche Lebensm Rundsch1994;90:143–6.¨[19]Kaufmann P,Herslof BG.A multivariate identification of natural triglyceride oils.Fat Sci Technol1991;93:179–83.´´[20]Forgacs E,Cserhati T.Charge-transfer chromatographic study of the interaction of antibiotics withsodium dodecylsulfate.J Pharm Biomed Anal1997;15:1295–302.[21]Biacs PA,Daood HG,Pavisa A,Hajdu F.Studies on the carotenoid pigments of paprika(Capsicumannuum L.var Sz-20).J Agric Food Chem1989;37:350–3.´´[22]Cserhati T,Forgacs E,Kiss V.High performance liquid chromatographic detection of a strongly retainedpigment fraction in Capsicum annuum.Die Nahrung1995;39:269–74.[23]O’Neil CA,Schwartz SJ.Photoisomerisation of b-carotene by photosensitization with chlorophylssderivatives as sensitizers.J Agric Food Chem1995;43:631–5.[24]Tonucci LH,Holden M,Beecher GR,Khachik F,Davis CS,Mulokozi G.Carotenoid content ofthermally processed tomato-based products.J Agric Food Chem1995;43:579–86.[25]Bueno MP.Collaborative study:determination of retinol and carotene by high-performance liquidchromatography.Food Chem1997;59:165–70.[26]Rozylo JK,Jaroniec M.Application of chromatographic and adsorption data for characterizing liquid/solid adsorption systems.Fresenius J Anal Chem1985;321:371–3.[27]Rozylo JK,Janicka M.Thermodynamic description of liquid–solid chromatography process in theoptimization of separation conditions of organic compound mixture.J Liq Chromatogr1991;14:3197–212.。
近年高考英语二轮复习 高考Ⅱ卷规范训练2(2021年整理)
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高考Ⅱ卷规范训练（二)Ⅰ.语法填空My name is Wil Wheaton—and I am a nerd（怪人）。
It's awesome to be 1.a nerd. When I was a little boy, people really teased me about that and 2。
made(make) me feel like there was something wrong with me for loving strange things。
Now that I'm an adult, I’m a 3.professional（profession） nerd, and the world has changed. I think we 4.have_realized(realize） that being a nerd is not about what you love 5。
but about how you love。
So there’s going to be a thing in your life that you love. I don’t know what it's going to be。
英语二作文模板零基础
Fourth, in addition to using transitional words, it is also important to use a variety of sentence structures in your writing. This can make your essay more engaging and enjoyable to read. For example, you can use simple sentences, compound sentences, and complex sentences to add variety to your writing.
writing 2
c) He is industrious. He is clever. d) The time for discussion was limited.
It was still possible to produce some interesting arguments.
D. To put it more simple, she is very slow at learning.
1. Susan speaks English like a nativespeaker.____.
2. She is taking a long time to improve her English.______.
Tasks
“Do you want another piece of toast, dear?” asked Mary. “No, thank you, I would like some more tea,” said Jim, holding out his cup. The Russells always had tea for breakfast, unlike most American families, who prefer coffee. Jim enjoys coffee much, he like tea, too, and make a pot of coffee for himself alone, he drinks tea with his wife. He says he gets all the coffee he wants at his office and at lunch downtown, he does not miss it at home.
英语作文unit2
英语作文unit2### Unit 2: The Power of CommunicationCommunication is the lifeblood of human interaction, and the mastery of English as a global language opens up a world of opportunities for effective communication. In this unit, we will explore the various facets of English communication, including verbal, non-verbal, and written forms.Verbal CommunicationVerbal communication is the most direct form of interaction. It involves speaking and listening, which are fundamental skills for expressing thoughts and understanding others. To excel in verbal communication, one must develop a rich vocabulary, clear pronunciation, and the ability to construct coherent sentences. Additionally, the art of conversation involves active listening and the ability to respond appropriately.Non-Verbal CommunicationNon-verbal communication encompasses a wide range of cues such as body language, facial expressions, gestures, and eye contact. These elements can convey emotions and attitudesthat words alone cannot express. For instance, a smile can indicate friendliness, while crossed arms might suggest defensiveness or disagreement. Understanding and utilizingnon-verbal cues effectively can greatly enhance the clarity and impact of one's message.Written CommunicationWritten communication is a crucial skill for expressing thoughts in a permanent and structured form. This includes writing emails, letters, reports, and essays. Good written communication requires not only a command of grammar andsyntax but also the ability to organize ideas logically and persuasively. In academic and professional settings, clearand concise writing is highly valued.Cultural ConsiderationsWhen communicating in English, it's important to be aware of cultural differences that can influence communication styles. For example, some cultures value directness and assertiveness, while others may prefer a more indirect and polite approach. Being sensitive to these differences can help avoid misunderstandings and foster positive relationships.ConclusionMastering the art of communication in English is a journeythat requires continuous learning and practice. By developing verbal, non-verbal, and written skills, and being mindful of cultural nuances, one can effectively navigate the global landscape and build meaningful connections with people from diverse backgrounds.Exercises1. Practice a conversation with a friend, focusing on using clear and concise language.2. Observe a public speaker and note the non-verbal cues they use to engage their audience.3. Write a short essay on a topic of your choice, ensuring that your writing is well-structured and free of grammatical errors.4. Research cultural communication styles of a country where English is commonly spoken and discuss how these styles might affect international interactions.This unit aims to provide a comprehensive understanding of the power of communication in English, equipping learners with the skills necessary to communicate effectively in various settings.。
八年级下英语u2的范文
八年级下英语u2的范文Learning a new language can be a challenging yet rewarding experience. As students in grade 8 English, we are currently exploring Unit 2 which focuses on a variety of topics and skills related to language acquisition. Through this unit, we have the opportunity to develop a deeper understanding of the English language and enhance our overall communication abilities.One of the key aspects of Unit 2 is vocabulary development. We are introduced to a wide range of new words and expressions that are essential for effective communication in various contexts. From academic and professional vocabulary to colloquial and idiomatic language, we are expanding our lexical repertoire. This not only helps us express ourselves more precisely but also allows us to comprehend and engage with a diverse range of texts and conversations.In addition to vocabulary, Unit 2 also emphasizes the importance of grammar and sentence structure. We delve into the intricacies of English grammar, learning about parts of speech, verb tenses,sentence patterns, and more. By mastering these grammatical concepts, we can construct more sophisticated and nuanced sentences, which in turn enhances our overall writing and speaking abilities. This knowledge also enables us to better understand the structure and flow of the English language, making it easier to comprehend complex texts and engage in meaningful discussions.Another key aspect of Unit 2 is the development of reading and listening comprehension skills. We are exposed to a variety of authentic materials, including articles, essays, podcasts, and videos, which challenge us to analyze and interpret the information presented. Through these activities, we learn to identify main ideas, understand context, and draw inferences. This not only strengthens our language proficiency but also fosters critical thinking and problem-solving skills, which are essential for academic and professional success.Furthermore, Unit 2 emphasizes the importance of effective communication, both in written and oral forms. We practice writing essays, reports, and other forms of academic writing, focusing on organization, coherence, and the use of appropriate language and tone. Additionally, we engage in various speaking activities, such as presentations, debates, and group discussions, which allow us to develop our public speaking skills and improve our confidence in expressing our ideas.One of the most valuable aspects of Unit 2 is the opportunity to explore cultural diversity and its influence on language. We delve into the relationship between language and culture, examining how cultural norms, traditions, and perspectives shape the way we communicate. This knowledge not only enhances our understanding of the English language but also fosters a greater appreciation for the diversity of human experiences and perspectives.As we progress through Unit 2, we also have the chance to develop essential study skills and learning strategies. From effective note-taking and time management to collaborative learning and self-reflection, these skills are crucial for our academic success and personal growth. By mastering these strategies, we can become more independent and self-directed learners, better equipped to navigate the challenges of language acquisition and academic achievement.In conclusion, Unit 2 of grade 8 English is a comprehensive and multifaceted learning experience that provides us with the tools and knowledge necessary to become proficient in the English language. Through the development of vocabulary, grammar, reading and listening comprehension, communication skills, and cultural awareness, we are gaining a deeper understanding of the complexities and nuances of the English language. As we continue toengage with the content and activities of this unit, we are not only improving our language skills but also cultivating critical thinking, problem-solving, and self-directed learning abilities – all of which will serve us well in our academic and personal endeavors.。
初二下册英语unit2作文
初二下册英语unit2作文# The Benefits of Learning a Second LanguageIn today's globalized world, learning a second language has become an essential skill for students. As a second-year middle school student, you are at a perfect age to start exploring the world of languages beyond your native tongue. This essay will discuss the numerous benefits of learning a second language, including cognitive development, cultural understanding, and enhanced career opportunities. Embracing a new language not only opens doors to new cultures but also significantly enriches your personal and professional life.## Cognitive DevelopmentOne of the most significant benefits of learning a second language is the positive impact it has on cognitive development. Studies have shown that bilingual individuals tend to have better problem-solving skills, enhanced memory, and improvedmultitasking abilities. When you learn a new language, your brain is constantly challenged to recognize and understand new sounds, words, and grammatical structures. This mental exercise strengthens the brain's executive functions, which are responsible for planning, decision-making, and controlling attention.Furthermore, learning a second language can delay the onset of age-related cognitive decline. Research indicates that bilingualism can contribute to maintaining cognitive functionsin old age, reducing the risk of Alzheimer's disease and other forms of dementia. Therefore, starting to learn a new language at a young age can have long-lasting benefits for your brain health.## Cultural UnderstandingAnother crucial advantage of learning a second language is the increased cultural understanding it provides. Language is deeply intertwined with culture, and by learning a new language,you gain insights into the traditions, values, and ways of life of the people who speak that language. This cultural awareness fosters empathy and broadens your perspective, allowing you to appreciate and respect diversity.For example, if you learn English as a second language, you can explore the rich literary and artistic heritage of English-speaking countries. You can read classic literature, watch films, and listen to music in their original language, gaining a deeper appreciation for these works. Additionally, understanding cultural nuances can enhance your ability to communicate effectively with people from different backgrounds, making you a more open-minded and globally aware individual.## Enhanced Career OpportunitiesIn today's competitive job market, proficiency in a second language can give you a significant edge. Employers increasingly value bilingual or multilingual employees, as they can communicate with international clients and partners,expand business opportunities, and navigate cross-cultural interactions more effectively. Learning a second language can thus open up a wide range of career possibilities in various fields such as international business, diplomacy, tourism, and translation.For instance, being fluent in English can be particularly advantageous. English is the most widely spoken language in the world and is often considered the global lingua franca. It is the primary language of international business, science, technology, and aviation. Mastering English can enable you to work in multinational companies, participate in international conferences, and access a vast amount of information and research that is predominantly available in English.## Personal Growth and ConfidenceLearning a second language also contributes to personal growth and boosts self-confidence. The process of acquiring a new language involves stepping out of your comfort zone,making mistakes, and overcoming challenges. This experience builds resilience and perseverance, qualities that are valuable in all aspects of life. As you progress and achieve fluency, you gain a sense of accomplishment and confidence in your abilities.Moreover, being bilingual can enhance your travel experiences. When you visit a country where you speak the language, you can interact with locals more meaningfully, understand signs and menus, and immerse yourself in the culture. This ability to connect with people from different parts of the world enriches your travel experiences and makes them more memorable and enjoyable.## Academic AchievementResearch has shown that students who learn a second language tend to perform better academically. Bilingual students often have higher scores in standardized tests, particularly in reading and math. The skills developed through language learning, such as critical thinking, problem-solving,and memory retention, transfer to other academic subjects, enhancing overall academic performance.Furthermore, learning a second language can improve your first language skills. Understanding the grammar and structure of another language can make you more aware of the intricacies of your native language. This heightened linguistic awareness can lead to better writing, reading, and communication skills in your first language.## ConclusionIn conclusion, learning a second language offers a multitude of benefits that extend beyond the ability to communicate in another tongue. It enhances cognitive development, fosters cultural understanding, opens up career opportunities, contributes to personal growth, and boosts academic achievement. As a second-year middle school student, you have a unique opportunity to start learning a new language and reap these benefits. Embrace the challenge, and you willfind that the rewards are well worth the effort. By becoming proficient in a second language, you will not only expand your horizons but also enrich your life in countless ways.。
Compulsory Course 2 of the English General Review
Influencing Cultural Trends
Popular film and television programs can influence fashion, music, and other cultural trends
Providing ainment
Film and television are important sources of entertainment and recovery for people of all ages
Social culture: Films and TV programs focus on various social phenotypes, customs, and lifestyles, promoting a window into different social cultures and enabling viewers to broaden their horizons
Promotion of artistic innovation
To familiarize students with the format and requirements of the College Entrance Examination, and to provide them with strategies and tips for effective exam preparation
01
Introduction
Chapter
Review purpose and requirements
To improve students' listening, speaking, reading, and writing skills through a variety of activities and exercises designed to enhance their language proficiency
高中英语必修2第二单元作文范文
高中英语必修2第二单元作文范文Here is an essay on the topic of "Sample Essay for Unit 2 of Compulsory English 2 in High School" with over 1000 words, written entirely in English without any additional punctuation marks.Education is a fundamental aspect of human development and plays a crucial role in shaping the future of individuals and societies. In the realm of high school education, the study of English as a second language holds immense significance as it not only enhances communication skills but also broadens cultural understanding and perspectives. The second unit of Compulsory English 2 in high school curriculum often focuses on various themes and topics that provide students with a deeper appreciation for the English language and its applications in the real world.One of the key themes explored in this unit is the concept of cultural diversity and its impact on global communication. As the world becomes increasingly interconnected, the ability to navigate and understand diverse cultural contexts has become essential. Through the study of this unit, students are exposed to the rich tapestry of cultural traditions, customs, and perspectives that exist across the globe. They learn to appreciate the nuances of language andcommunication that arise from these cultural differences, and develop the skills to effectively bridge the gaps and foster cross-cultural understanding.Another important aspect of this unit is the emphasis on practical language skills and their application in real-world scenarios. Students are often presented with case studies or role-playing exercises that simulate situations where they must utilize their English language proficiency to engage in meaningful dialogues, negotiate solutions, and effectively communicate their ideas. This hands-on approach not only reinforces the theoretical concepts learned in the classroom but also equips students with the confidence and competence to navigate the challenges of the modern workplace and global marketplace.Moreover, the second unit of Compulsory English 2 often delves into the realm of academic writing and research. Students are introduced to the conventions and structures of academic essays, research papers, and other forms of scholarly discourse. They learn to critically analyze sources, synthesize information, and articulate their own ideas and arguments in a clear and coherent manner. This skill set is not only valuable for their academic pursuits but also prepares them for the rigors of higher education and professional writing.Beyond the specific content and skills covered in this unit, the studyof Compulsory English 2 also plays a crucial role in developing students' overall language proficiency and cognitive abilities. The process of learning a second language has been shown to enhance critical thinking, problem-solving, and cognitive flexibility. As students navigate the complexities of grammar, vocabulary, and cultural nuances, they cultivate a deeper understanding of language structures and the ability to think in multiple perspectives.Furthermore, the study of Compulsory English 2 serves as a gateway to a world of opportunities. In an increasingly globalized economy, the ability to communicate effectively in English is a highly sought-after skill that opens doors to a wide range of academic, professional, and personal pursuits. Whether it is pursuing higher education in an English-speaking institution, securing employment in multinational corporations, or engaging in international travel and cultural exchange, the proficiency gained through this unit can significantly enhance students' future prospects and personal growth.In conclusion, the second unit of Compulsory English 2 in high school curriculum is a vital component of the educational experience that prepares students for the challenges and opportunities of the21st century. Through the exploration of cultural diversity, practical language skills, and academic writing, students develop a comprehensive understanding of the English language and its role in shaping global communication and intercultural exchange. As theynavigate the complexities of this unit, they not only enhance their linguistic abilities but also cultivate essential cognitive and critical thinking skills that will serve them well throughout their academic and professional journeys. The significance of this unit cannot be overstated, as it lays the foundation for a lifetime of personal and professional growth, fueled by the power of the English language.。
高中英语必修2第二单元作文范文
高中英语必修2第二单元作文范文The Importance of Effective Communication in the Modern WorldIn today's rapidly evolving global landscape, the ability to communicate effectively has become increasingly crucial. As we navigate the complexities of personal, academic, and professional realms, the skills to convey ideas, exchange information, and build meaningful connections have become indispensable. This essay will delve into the significance of effective communication, particularly in the context of the modern world, and explore strategies for developing and honing these essential abilities.Firstly, effective communication is the cornerstone of successful interpersonal relationships. Whether we are interacting with family members, friends, colleagues, or strangers, the capacity to listen attentively, express our thoughts and feelings clearly, and engage in constructive dialogue is paramount. Strong communication skills enable us to foster deeper connections, resolve conflicts, and cultivate mutual understanding. In the personal sphere, effective communication can strengthen bonds, enhance empathy, and leadto more fulfilling relationships.Beyond the personal realm, effective communication is also vital in the academic and professional domains. In the classroom, the ability to articulate ideas, ask thoughtful questions, and participate in discussions can significantly contribute to academic success. Students who can effectively communicate their understanding of course material, engage in collaborative learning, and express their perspectives are more likely to excel. Similarly, in the workplace, the capacity to communicate effectively with colleagues, clients, and superiors is a crucial asset. Professionals who can present their ideas persuasively, write concise and clear reports, and engage in productive team discussions are often more successful in their careers and contribute more effectively to their organizations.Moreover, effective communication is essential for navigating the increasingly globalized world. As we interact with individuals from diverse cultural backgrounds, the ability to adapt our communication styles, understand cultural nuances, and bridge linguistic barriers becomes crucial. Effective cross-cultural communication fosters mutual understanding, facilitates the exchange of ideas, and enables successful collaboration on a global scale. In an interconnected world, the capacity to communicate effectively across cultures is a valuable skill that can open doors to new opportunities and foster greater international cooperation.In addition to the practical benefits, effective communication also plays a crucial role in personal growth and self-expression. The ability to articulate our thoughts, feelings, and experiences can be a powerful tool for self-reflection, emotional processing, and personal development. Effective communication allows us to share our unique perspectives, advocate for our needs and beliefs, and contribute meaningfully to societal discourse. This, in turn, can enhance our sense of agency, confidence, and overall well-being.To develop and hone effective communication skills, individuals can employ various strategies. Firstly, active listening is a fundamental skill that involves fully engaging with the speaker, seeking to understand their message, and responding thoughtfully. By actively listening, we can gather valuable information, demonstrate respect, and foster more constructive dialogues. Additionally, practicing clear and concise expression, both in written and oral forms, is essential. This includes developing the ability to organize our thoughts, use appropriate language, and tailor our communication style to the audience and context.Furthermore, developing empathy and cultural awareness can greatly enhance our communication skills. By striving to understand different perspectives, acknowledging cultural differences, and adapting our communication approach accordingly, we can buildstronger connections and facilitate more effective exchanges. Lastly, seeking feedback and continuously reflecting on our communication patterns can help us identify areas for improvement and refine our skills over time.In conclusion, effective communication is a vital skill in the modern world, with far-reaching implications for our personal, academic, and professional lives. By cultivating the ability to listen attentively, express ourselves clearly, and adapt our communication styles to diverse contexts, we can strengthen relationships, achieve academic and professional success, and contribute to a more interconnected global community. As we navigate the complexities of the 21st century, the importance of effective communication cannot be overstated, and investing in the development of these skills can lead to a more fulfilling and impactful life.。
高中英语选修二unit2作文
高中英语选修二unit2作文As a language model AI, I'm not able to generate essays for tests or exams. However, I can provide guidance on how to write an essay for this assignment.First of all, you may want to start by introducing the topic of the essay, which is likely related to the content of Unit 2 in your English textbook. Then, you can present your main points or arguments in a clear and organized manner. It's important to support your points with relevant examples and evidence from the text or other sources. Additionally, you should consider addressing any potential counterarguments to strengthen your position.In the conclusion, you should summarize the key points you've made and restate your main argument. It's also a good idea to leave the reader with some food for thought or a thought-provoking question.Now, let's switch to Chinese.在中文部分，你可以进一步阐述你对这个主题的看法，以及你对课文中所涉及的内容的理解。
如何写第二单元作文英语
如何写第二单元作文英语How to Write an English Essay on Unit 2。
Writing an English essay on Unit 2 can be a challenging task for many students. However, with the right approach,it can be a rewarding experience that helps to improve your writing skills. In this article, we will provide you with some tips on how to write an excellent essay on Unit 2.1. Understand the Topic。
The first step in writing an essay on Unit 2 is to understand the topic. Read the question carefully and make sure you know what it is asking you to do. If you are unsure, ask your teacher for clarification.2. Research the Topic。
Once you understand the topic, it is time to research it. Look for relevant information in textbooks, onlineresources, and academic journals. Take notes as you readand make sure you cite your sources properly.3. Plan Your Essay。
第二单元作文写读后感,内容略写
第二单元作文写读后感,内容略写下载温馨提示:该文档是我店铺精心编制而成，希望大家下载以后，能够帮助大家解决实际的问题。
文档下载后可定制随意修改，请根据实际需要进行相应的调整和使用，谢谢!并且，本店铺为大家提供各种各样类型的实用资料，如教育随笔、日记赏析、句子摘抄、古诗大全、经典美文、话题作文、工作总结、词语解析、文案摘录、其他资料等等，如想了解不同资料格式和写法，敬请关注!Download tips: This document is carefully compiled by theeditor. I hope that after you download them,they can help yousolve practical problems. The document can be customized andmodified after downloading,please adjust and use it according toactual needs, thank you!In addition, our shop provides you with various types ofpractical materials,such as educational essays, diaryappreciation,sentence excerpts,ancient poems,classic articles,topic composition,work summary,word parsing,copy excerpts,other materials and so on,want to know different data formats andwriting methods,please pay attention!《读〈鲁滨逊漂流记〉有感》。
读完《鲁滨逊漂流记》这本书，我的心里就像有一个小人在不停地说话。
English Writng 2
English Argumentative Writing
Instructed by: Mr. LIU Xiao Kai
What is an INTRODUCTION paragraph? What should contains in an INTRODUCTION paragraph?
English Argumentative Writing
English Argumentative Writing
Instructed by: Mr. LIU Xiao Kai
“Moves” that an introduction paragraph follows:
English Argumentative Writing
Instructed by: Mr. LIU Xiao Kai
Instructed by: Mr. LIU Xiao Kai
What is an INTRODUCTION paragraph?
The first paragraph (usually one or two) It is to serve as a 'map' of the essay Should be written clearly
An introduction paragraph may contain the following:
Relevant background information
A thesis statement (Your point of view)
Overview of the thesis
Broad statement (evaluate, forecast future, make recommendations)
0基础英语2作文模板
0基础英语2作文模板Title: English Writing Template for Beginners。
As a beginner in learning English, it can be challenging to write a well-structured and coherent essay. However, with the help of a simple template, you can easily organize your thoughts and ideas to create a clear and effective piece of writing. In this article, we will provide you with a basic English writing template that you can use as a guide to improve your writing skills.Introduction:The introduction is the opening paragraph of your essay, and it should provide a brief overview of the topic you will be discussing. It should also include a thesis statement, which is a concise summary of the main point or argument of your essay. For example, if the topic of your essay is "The Importance of Exercise," your introduction could start with a general statement about the benefits of staying active, and then lead into your thesis statement, such as "Regular exercise is essential for maintaining a healthy lifestyle and preventing various health problems."Body Paragraphs:The body of your essay should consist of several paragraphs that expand on the main points of your thesis statement. Each paragraph should focus on a single idea or topic, and they should be logically organized to provide a smooth transition from one point to the next. For example, if you are writing about the benefits of exercise, you could have separate paragraphs for physical health, mental well-being, and social benefits. Each paragraph should start with a topic sentence that introduces the main idea of the paragraph, followed by supporting details and examples to strengthen your argument.Conclusion:The conclusion is the final paragraph of your essay, and it should summarize the main points you have made in the body of the essay. It should also restate your thesisstatement in different words and provide a sense of closure to your essay. Additionally, you can include a call to action or a thought-provoking statement to leave a lasting impression on your readers. For example, if you are writing about the importance of exercise, your conclusion could encourage readers to make physical activity a part of their daily routine for a healthier and happier life.Sample Essay Using the Template:Topic: The Benefits of Reading。
composition 2
I went to deliver to this afternoon. It was rainy when I arrived at Li Village. I was afraid that those letters would be soaked by rain, so I took off my coat and covered the letters with it. It was raining heavily and heavily. I felt a little cold, but I could stop the delivering, because I knew there were a lot of people were waiting for the letters. Ispoke to myself “ come on !” When I arrived at Sun Village, I found a girl who wore a red raincoat was waiting for me outside her house. I found out her letter and gave it to her. Then I was on my way. Suddenly, I heard the little girl call me back. So I stopped. She took off her raincoat and gave it to me., “ Take this, bye!” And then she ran away quickly. I didn’t feel cold at all at that time. I knew if I helped the other people, they would help me too.臧晓辉 臧晓辉
高中第二册第二单元作文：我难忘的一件事作文字高中第二册第二单元作文
高中第二册第二单元作文：我难忘的一件事
在我生命中的很多时刻，有一次经历至今仍历历在目，无法忘怀。
那是我在高中时期发生的一件事。
背景
那时候我正处在高中的第二年，成绩平平，生活麻木而平凡。
一天，班里的老师宣布了一个好消息，我们班被选中参加一项国家级的科技竞赛。
我觉得这可能是我改变生活的机会，于是我决定全力以赴。
准备
为了参加这项科技竞赛，我们必须组建一个有潜力的团队。
我很幸运地加入了其中，并被任命为主要的科技团队领袖。
我很快开始了我的工作，收集资料、制定计划、分配任务等等。
我尝试通过各种方法来提高我们团队的合作效率，如定期进度汇报和互相学习。
我们每个人都非常努力地工作着，但是随着时间的推进，我们发现需要一些更高端的技术来确保我们项目的成功。
竞赛
到了竞赛的那天，我和我的同伴们紧张而期待着。
我的脑子里充满了各种奇怪的想法，我甚至觉得自己会在最关键的时候出错。
但是，当我和我的队友们开始展示我们的项目时，我突然感到自己的自信心增强了。
我们的项目非常成功，让评审团和观众们全都投入其中。
他们被我们的创意和实现所震撼，而我们团队的成员们也被我们自己的表现所鼓舞。
总结
这是我人生中的一次难忘的经历，因为它让我看到了我更大的潜力。
通过这个科技竞赛，我学到了很多新技术，从而成为了一名更加优秀的科技工程师。
我也学到了团队合作的重要性，尤其是在科技领域这一点更重要。
我坚信，如果我们能够以正确的方式使用技术和合作精神，我们可以创造出许多令人惊叹的事物。
我不会忘记这个经历，它激励我做出更大的贡献。
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聪明的汽车摘要为了解决停车空位较短时，路边侧位停车困难这一问题，我们先进行了资料查阅和分析，得出以下结论：在侧位停车的过程中，汽车动力学特性及环境因素对停车的成功与否影响最大。
汽车的动力学特性即车辆自身参数，包括车长、车宽、前悬、最小转弯半径；环境因素主要是指道路参数，包括停车位长度、宽度、停车初始位置和角度。
考虑到问题的简化，我们只针对特定一种车型，从由简到繁的停车入位步骤入手，主要考虑环境因素对停车的影响，以解决问题。
首先，我们通过建立平面直角坐标系，将停车这一空间问题转化为二维平面问题，之后，我们把该车顺利停入停车位的过程分为了n(n=1,2,3,4,…)个步骤，再根据数学归纳法，对该车入位所需的步骤个数进行讨论分析：1．假设该车可以经过两个步骤顺利停入停车位。
结合图形列式，得到θi和r i(i=1,2)的关系，运用控制变量法，求解θit(i=1,2)，再对θit进行反复修正，从而得到模型一。
2. 假设该车经过三个步骤才能停入停车位。
这是因为停车位位宽不确定，该车可能经过两个步骤后却不能停入停车位，因此我们增加了一个停车步骤。
通过列出转角θi(i=1,2,3)和转弯半径ri(i=1,2,3)之间的关系，我们建立出模型二。
由于模型二是在模型一的基础上建立的，因此，模型二可以用与模型一相同的方法进行问题求解。
3. 若该车经过三个步骤仍不能顺利停入停车位，我们则假设经过n个步骤后，该车方能顺利停入停车位。
于是我们应用与模型一、模型二相同的方法建立模型，求得通解。
模型三即可概括该车停入停车位的每个步骤所对应的转弯半径和角度，从而可得到该车应从哪个角度、哪个位置进入停车位，从而问题得到解决。
通过以上模型的建立及应用matlab软件求解分析，我们得出结论：只要停车位的位长大于该车车长，无论两者的差值有多小，该车都能顺利停入车位，只是停入车位的步骤个数会有所不同。
关键词：侧位停车转弯半径转弯角度角度修正法一、问题重述在狭窄的空间里把车停放在合适的位置，或在短小的停车位上侧位停车，一直是考验驾驶员技术与信心的问题。
有调查报告称：57% 的驾驶员对自己的停车技术缺乏自信，这一方面影响人的驾驶体验，一方面也使停车空间不能得到充分利用。
在此，我们协助驾驶员解决停车的问题：(1) 对侧位停车而言，在空位较短的时候，驾驶员会难以确定自己的汽车是否能顺利停入。
我们通过建立合理的模型，来判断此车是否能在该处侧位停车。
(2) 由模型假设，我们给出本车为了进入停车位，应当从何位置及角度进入，并在图中显示出理想线路以及允许的偏差。
二、模型假设1．建立模型时选用丰田卡罗拉1.6 GL AT为模型车，是由于其几何尺寸、转弯半径等容易测得，且该车型位于2009年全球畅销汽车排行榜榜首，极具代表性，下文中将简称为模型车。
2．假设可以得到包括停车空位的长度宽度等数据的停车位置的平面图，且该平面图能够显示在汽车的车载显示器上。
3．确定初始位置的条件为：(1)停车过程中方向盘打死时，车身和前一车位不发生任何干涉；(2)假设路旁停车位为一个停车库，停车过程中车身不能和车库的库底发生碰撞，车轮不能和库边发生碰撞。
4．当模型中有符号的使用说明时以模型中符号的使用说明为准。
三、符号说明四、数据分析一）汽车相关参数的描述 如图1.1和图1.2所示。
1.车长：沿汽车的前进方向最前端到后端的距离。
2.车宽：汽车最左端到最右端的距离，不包括反光镜、转向侧灯、脚踏板等部位。
3.轴距：汽车前轴中心至后轴中心的距离。
4.前悬：汽车最前端至前轴中心的距离。
汽车的基本参数中不包括前悬，需通过车长与轴距计算。
5.最小转弯半径：转向盘转到极限位置时的转弯半径。
转弯半径越小，汽车的机动性能越好。
如图1.3所示，r 为汽车的最小转弯半径。
6．车位长度：路旁停车位前后都停有车辆时，前方车辆尾部到后方车辆头部的距离。
模型车的基本数据 [1]如下所示： 车长l/宽w(mm)：4540/1760 轴距zj （mm ）：2600最小转弯半径r(mm)：5200 前悬qx （mm ）：(l-zj)/2=470二）停车运动的平面简化1．图2.11、2.12、2.13为停车初始时，车身与水平方向的夹角θ变化时的运动情况。
2．车轮运动轨迹的优化在研究侧位停车之前，首要需明确停车时车轮的运动轨迹。
因此需要建立车辆停车的运动学模型，以推导模型车的运动轨迹方程式，为之后的模型建立基础。
以下为停车运动轨迹方程式的具体推导过程。
其中(xf,yf)为前轴中心轴坐标，(xr,yr)为后轴中心点坐标，vl为轴距，w为后轮距，Φ为前轴中心转向角，θ为车辆中心轴与水平方向的夹角。
正常情况下，停车的速度很低，一般不大于5km/h ，由此假定无滑轮的现象，后轮轨迹的垂直方向的速度为0，其方程的表达式如下：前后轴的中心点坐标关系：（2.21）而由图2.2可知，前后轴中心点坐标关系为：（2.22）对(2.22)式积分，可得其速度关系为：（2.23）将(2.23)式代入(2.21)式中可得关系式：（2.24）前后中心点的x ，y 方向速度为：（2.25）由此将(2.25)式代入(2.24)式，即可求得车辆回转圆角速度为：（2.26）将(2.25)和(2.26)代入(2.23)式中，即可求得后轮中心点x ，y 方向速度分别为：（2.27）最后将(2.26)式对时间积分并代入(2.27)式后，再对时间积分即可求得后轴中心点的轨迹方程式：（2.28）根据图2.2中坐标位置的关系，可进一步求出左右轮中心点的运动轨迹方程式：左轮：（2.29）右轮：（2.29’）由（2.29）式和（2.29’）式知，当低速停车时，且不考虑轮胎侧滑时，后轮的行径轨迹与轮距，轴距及前轴中心点转向角有关，而与停车速度无关。
三）两平行车辆间的水平距离对模型的影响具体到某一次停车，车长、车宽、车位宽、最小转弯半径为定值时，两平行车辆水平距离对最小停车位长度影响最大，性作用。
令两平行车辆间水平距离为d，其初值为d=0.3m，依次增大d，通过数据采集[2]，可得到d不断变化时对应的泊车转向角与两次泊入时最小停车位的长度，如表3.1所示。
其中，车位宽度为固定值2.2m，泊车结束时车辆泊在停车位的正中央。
根据以上数据制图，由图3.2可看出，当水平距离d逐渐增大时，停车转向角及最小停车位长度的变化势大致相同，都遵循先减小再增大的变化趋势。
当水平距离d从300mm逐渐增大到900mm时，停车转向角及最小停车位长度的值逐渐减小，直到极限最小值；随着d从900mm继续增大，停车转向角及最小停车位长度也随之增大。
对模型车而言，d=900mm为停车最佳初始位置，此时停车转向角=44°，“两次停车”可完成停车时最小停车位长度lw=6546mm，此水平距离d能使试验车实现最快最佳泊车。
不同型号的车辆停车时有不同的最佳初始位置，结合专家经验及数据验证，两平行车辆间水平最佳距离大都集中在1米左右(大约一车门距离)，泊车转向角集中在45°附近。
这个数据可以作为普通车辆停车时的参考距离。
五、模型的建立与求解一）停车过程步骤化1．停车过程是后车轮改变方向反复画圆弧的过程，但圆弧半径与弧度要受到停车位空间和车辆参数的约束。
我们令停车时后车轮轨迹顺时针为正，逆时针为负，停车轨迹改变一次方向为停车的一个步骤，由此停车过程分为了n步骤（n=1,2,3,4,…）,设j个步骤可成功停车，j≤n,（j=1,2,3,4,…）。
图1—图4所示为停车的4个步骤。
2．在停车模型的建立前，我们需要把抽象、复杂的实际问题过渡到具体、直接的符号运算之中，则有如下计算：设车辆右车轮中心点横坐标值为x rn，左下角纵坐标值为Y ln。
车长l、车宽w、前悬qx、最小转弯半径r、模型车a与平行车辆b的水平距离d已知，车位宽cw按道路标准设为定值2.2m。
当车辆停在停车位正中央时，车辆与停车位右侧水平最小距离为wx，车辆右后轮中心点与停车位右侧水平距离为dj n。
由图1得：由车辆停车的运动学模型可知，车辆停车时车轮运动轨迹为圆，模型车a要停入停车位，停车轨迹由参数r1，r2，r3…r n及θ1,θ2,θ3…θn决定。
因此，对于一停车任务，如何停车就转化为如何求未知参数r n与θn的值，可以用以下方程式来表示：(1)(1)式则等价于(2)式：(2)其中，分析图1—图4并结合实际可知，车辆，至少需要两次才能停入车位。
因此，我们建立以下三个模型进行具体分析。
二）建立模型◆．模型一：两次停入停车位1．侧位停车系统模型的建立停车一个步骤后，记后右轮中心点为a(a x,a y)，车头右端点为a1(a1x,a1y)，后左轮中心点为a2(a2x,a2y)，车尾左端点为ha2(ha2x,ha2y) 。
容易得出：停车两个步骤后，记后右轮中心点为b(b x,b y)，车头右端点为b1(b1x,b1y)，后左轮中心点为b2(b2x,b2y)，车尾左端点记为hb2(hb2x,hb2y) 。
容易得出：2．侧位停车轨迹算法步骤1：假设r 2已知，求θ2t 的值 （1）令r n=r （n=1，2，3…），此时θ2t =0，整理式子可得： 由可求出θ2t 的值，代入中，此时y rk 为两平行车辆间距离为d 时的最小停车位车辆恰好停入停车位正中央。
（2）r 2 和 θ2t 的修正由于转弯半径和转弯角度受停车位长度和宽度的约束，因此要对r 2 和 θ2t 进行修正。
当停车位的距离大于或等于最小停车位时，车辆两个步骤后就停入停车位。
停车一个步骤后，车辆水平方向前进d 1，垂直方向前进v 1，停车两个步骤后，车辆水平方向前进d 2，垂直方向前进v 2，那么可以得到：(1.1)将（1.1）（1.2）带入（1.3）得：两个步骤后模型车停入停车位，这时θ2=0，即θ1=θ2。
令r 1=r 2=r ，可求得θ1。
对模型车而言，由模型假设第3点的(1)知，需要对θ1进行修正，下图为θ1的修正图：停车第二步骤完成后，取轨迹上的任意一点M(m x,m y)，车身与垂直方向的夹角为β，则M点坐标为：（1.4）M1(m1x,m1y)的坐标为：（1.5）将（1.4）和a(a x,b x)带入（1.5）得：①（1.6）②由于车身不能和前车发生空间干涉，那么，当r1=r2，m1x=w+d时，m1y需要满足：m1y≤—100（为了留够充足的缓冲距离100mm）。
将m1x=w+d代入（1.6）式的①式中，得：将m1y≤—100代入（1.6）式的②式，得：整理上面的式子可求出β；若不满足，则令θ2=θ1+1·π/180，重新整理求出β；若仍不满足，则循环前面的式子，求出恰好不会碰撞时θ1的最小值。
若车位足够长，则θ2t=0。