Effects of Vertex Activity and Self-organized Criticality Behavior on a Weighted Evolving Networ

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motivation in language learning tesol internation

MOTIVATION IN LANGUAGE LEARNING: SELECTED REFERENCES(last updated 9 May 2013)Ames, C. (1984). Competitive, cooperative, and individualistic goal structures: A cognitive-instructional analysis. In R. E. Ames & C. Ames (Eds.), Research on motivation ineducation (pp. 177-207). New York: Academic Press.Ames, C. (1986). Effective motivation: The contribution of the learning environment. In R. S.Feldman (Ed.), The social psychology of education (pp. 235-256). Cambridge:Cambridge University Press.Amrein, A., & Berliner, D. (2003). The effects of high-stakes testing on student motivation and learning. Educational Leadership, 60(5), 32-38.Bacon, S. M., & Finnemann, M. D. (1990). A study of the attitudes, motives, and strategies of university foreign language students and their disposition to authentic oral and writteninput. Modern Language Journal, 74, 459-473.Bandura, A., & Schunk, D. H. (1981). Cultivating competence, self-efficacy, and intrinsic interest through proximal self-motivation. Journal of Personality and Social Psychology, 41, 586-598.Benson, M. (1991). Attitudes and motivation towards English: A survey of Japanese freshmen.RELC Journal, 22(1), 34-48.Bråten, I., & Olaussen, B. S. (1998). The relationship between motivational beliefs and learning strategy use among Norwegian college students. Contemporary Educational Psychology, 23, 182-194.Brophy, J. (1998). Motivating students to learn. New York: McGraw-Hill.Brophy, J. E. (1999). Towards a model of the value aspects of motivation in education: Developing an appreciation for particular domains and activities. EducationalPsychologist, 34, 75-85.Brophy, J. & Kher, N. (1986). Teacher socialization as a mechanism for developing student motivation to learn. In R. S. Feldman (Ed.), The social psychology of education (pp. 257-288). Cambridge: Cambridge University Press.Brown, J. D., Cunha, M. I. A., & Frota, S. de F. N. (2001). The development and validation of a Portuguese version of the Motivated Strategies for Learning Questionnarie. In Z. Dörnyei & R. Schmidt (Eds.), Motivation and second language acquisition (pp. 257-280).Honolulu, HI: Second Language Teaching & Curriculum Center, University of Hawai‘i Press.Brown, S., Armstrong, S., & Thompson, G. (1998). Motivating students. London: Kogan Page.Chambers, G. (1999). Motivating language learners. Clevedon, UK: Multilingual Matters. Cheng, H-F., & Dörnyei, Z. (2007). The use of motivational strategies in language instruction: The case of EFL teaching in Taiwan. Innovation in language learning and teaching, 1,153-174.Cohen, M., & Dörnyei, Z. (2002). Focus on the language learner: Motivation, styles and strategies. In N. Schmidt (Ed.), An introduction to applied linguistics (pp. 170-190).London, UK: Arnold.Cooper, H., & Tom, D. Y. H. 1984. SES and ethnic differences in achievement motivation. In R.E. Ames & C. Ames (Eds.), Motivation in education (pp. 209-242). New York:Academic Press.Cranmer, D. (1996). Motivating high level learners. Harlow, UK: Longman.Crookes, G., & Schmidt, R. W. (1991). Motivation: Re-opening the research agenda. Language Learning, 41, 469-512.Csikszentmihalyi, M., & Nakamura, J. (1989). The dynamics of intrinsic motivation: A study of adolescents. In R. Ames & C. Ames (Eds.), Handbook of motivation theory and research, Vol. 3: Goals and cognitions (pp. 45–71). New York, NY: Academic Press.Csizér, K., & Dörnyei, Z. (2005). Language learners’ motivational profiles and their motivated learning behavior. Language Learning, 55, 613-659.Csizér, K., Kormos, J., & Sarkadi, Á. (2010). The dynamics of language learning attitudes and motivation: Lessons from an interview study of dyslexic language learners. ModernLanguage Journal, 94(3), 470-487.deCharms, R. (1984). Motivation enhancement in educational settings. In R. E. Ames & C.Ames (Eds.), Motivation in education (pp. 275-310). New York: Academic Press. Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behavior. New York, NY: Plenum.Deci, E. L., & Ryan, R. M. (1991). A motivational approach to self: Integration in personality. In R. A. Dienstbier (Ed.), Perspectives on motivation: Nebraska Symposium on Motivation, 1990 (pp. 237-288). Lincoln, NE: University of Nebraska Press.Deci, E. L., Vallerand, R. J., Pelletier, L. G., & Ryan, R. M. (1991). Motivation and education: The self-determination perspective. Educational Psychologist, 26(3 & 4), 325-346.De Volder, M. L., & Lens, W. (1982). Academic achievement and future time perspective as a cognitive-motivational concept. Journal of Personality and Social Psychology, 44, 20-33Dörnyei, Z. (1990). Conceptualizing motivation in foreign language learning. Language Learning, 40, 45-78.Dörnyei, Z. (1994). Motivation and motivating in the foreign language classroom. Modern Language Journal,78, 273-284.Dörnyei, Z. (1998). Motivation in second and foreign language learning. Language Teaching, 31, 117-135.Dörnyei, Z. (2001). Motivational strategies in the language classroom. Cambridge, UK: Cambridge University Press.Dörnyei, Z. (2001). Teaching and researching motivation. Harlow, UK: Longman.Dörnyei, Z. (2002). The motivational basis of language learning tasks. In P. Robinson (Ed.), Individual differences and instructed language learning (pp. 137-158). Philadelphia, PA: John Benjamins.Dörnyei, Z. (2003). Attitudes, orientations, and motivations in language learning: Advances in theory, research, and applications. Language Learning, 53(Supplement 1), 3-32.Dörnyei, Z. (2007). Creating a motivating classroom environment. In J. Cummins & C. Davison (Eds.), International Handbook of English Language Teaching (Vol. 2) (pp. 719-731).New York, NY: Springer.Dörnyei, Z. (2008). New ways of motivating foreign language learners: Generating vision. Links, 38(Winter), 3-4.Dörnyei, Z. (2009). The L2 motivational self system. In Z. Dörnyei & E. Ushioda (Eds.), Motivation, language identity and the L2 self (pp. 9-42).Tonawanda, NY: MultilingualMatters.Dörnyei, Z., & Csizér, K. (1998). Ten commandments for motivating language learners: Results of an empirical study. Language Teaching Research, 2, 203-229.Dörnyei, Z., & Csizér, K. (2002). Some dynamics of language attitudes and motivation: Results of a longitudinal nationwide survey. Applied Linguistics, 23, 421-462.Dörnyei, Z., & K. Csizér. (2002). Some dynamics of language attitudes and motivation: Results of a longitudinal nationwide survey. Applied Linguistics, 23, 421–462.Dörnyei, Z., Csizér, K., & Németh, N. (2006). Motivation, language attitudes, and globalization:A Hungarian perspective. Clevedon, UK: Multilingual Matters.Dörnyei, Z., & Ottó, I. (1998). Motivation in action: A process model of L2 motivation. Working Papers in Applied Linguistics, 4, 43-69.Dörnyei, Z., & Schmidt, R. (Eds.). (2001). Motivation and second language acquisition.Honolulu: National Foreign Language Resource Center/ University of Hawai'i Press.Dörnyei, Z., & Skehan, P. (2003). Individual differences in second language learning. In C. J.Doughty & M. H. Long (Eds.), The handbook of second language acquisition (pp. 589-630). Malden, MA: Blackwell.Dweck, C. S. (1986). Motivational processes affecting learning. American Psychologist, 41, 1040-1048.Ehrman, M. (1996). An exploration of adult language learner motivation, self-efficacy, and anxiety. In R. Oxford (Ed.), Language learning motivation: Pathways to the new century (pp. 81-103). Honolulu, HI: University of Hawai’i Press.Gao, Y., Zhao, Y., Cheng, Y., & Zhou, Y. (2004). Motivation types of Chinese university undergraduates. Asian Journal of English Language Teaching, 14, 45-64.Gardner, R. (1985). Social psychology and second language learning: The role of attitude and motivation. London, UK: Edward Arnold.Gardner, R. (2001). Integrative motivation and second language acquisition. In Z. Dörnyei & R.Schmidt (Eds.), Motivation and language acquisition (pp. 1-19). Honolulu, HI:University of Hawai’i Press.Gardner, R. C. (2000). Correlation, causation, motivation, and second language acquisition.Canadian Psychology, 41, 10-24.Gardner, R. C. (2001). Integrative motivation: Past, present and future. Retrieved from ://publish.uwo.ca/~gardner/docs/GardnerPublicLecture1.pdfGardner, R. (2001). Integrative motivation and second language acquisition. In Z. Dörnyei & R.Schmidt (Eds.), Motivation and language acquisition (pp. 1-19). Honolulu: University of Hawai’i Press.Gardner, R. C. (2005). Integrative motivation and second language acquisition. Retrieved from ://publish.uwo.ca/~gardner/docs/caaltalk5final.pdfGardner, R. C. (2009). Gardner and Lambert (1959): Fifty years and counting. Retrieved from ://publish.uwo.ca/~gardner/docs/CAALOttawa2009talkc.pdfGardner, R. C., & Lambert, W. E. (1959). Motivational variables in second language acquisition.Canadian Journal of Psychology, 13, 266-272.Gardner, R. C., & Lambert, W. E. (1972). Attitudes and motivation in second language learning. Rowley, MA: Newbury House.Gardner, R. C., Masgoret, A.-M., Tennant, J., & Mihic, L. (2004). Integrative motivation: Changes during a year-long intermediate-level language course. Language Learning, 54, 1-34.Gardner, R. C., & Smythe, P. C. (1981). On the development of the attitude/ motivation test battery. The Canadian Modern Language Review, 37, 510-525.Gardner, R. C., & Tremblay, P. F. (1994). On motivation, research agendas, and theoretical frameworks. The Modern Language Journal, 78, 359-368.Grabe, W. (2009). Motivation and reading. In W. Grabe (Ed.), Reading in a second language: Moving from theory to practice (pp. 175-193). New York, NY: Cambridge UniversityPress.Graham, S. (1994). Classroom motivation from an attitudinal perspective. In H. F. J. O'Neil and M. Drillings (Eds.), Motivation: Theory and research (pp. 31-48). Hillsdale, NJ:Lawrence Erlbaum.Guilloteaux, M. J., & Dörnyei, Z. (2008). Motivating language learners: A classroom-oriented investigation of the effects of motivational strategies on student motivation. TESOLQuarterly, 42, 55-77.Hadfield, J. (2013). A second self: Translating motivation theory into practice. In T. Pattison (Ed.), IATEFL 2012: Glasgow Conference Selections (pp. 44-47). Canterbury, UK:IATEFL.Hao, M., Liu, M., & Hao, R. (2004). An empirical study on anxiety and motivation in English asa Foreign Language. Asian Journal of English Language Teaching, 14, 89-104.Hara, C., & Sarver, W. T. (2010). Magic in ESL: An observation of student motivation in an ESL class. In G. Park, H. P. Widodo, & A. Cirocki (Eds.), Observation of teaching:Bridging theory and practice through research on teaching (pp. 141-153). Munich,Germany: LINCOM EUROPA.Hashimoto, Y. (2002). Motivation and willingness to communicate as predictors of reported L2 use: The Japanese ESL context. Retrieved from :///sls/uhwpesl/20(2)/Hashimoto.pdf.Hawkins, J. N. (1994). Issues of motivation in Asian education. In H. F. O’Neill & M. Drillings (Eds.), Motivation—theory and research (pp. 101-115). Hillsdale, NJ: Erlbaum. Hsieh, P. (2008). Why are college foreign language students' self-efficacy, attitude, and motivation so different? International Education, 38(1), 76-94.Huang, S. (2010). Convergent vs. divergent assessment: Impact on college EFL students’ motivation and self-regulated learning strategies. Language Testing, 28(2), 251-270.Keller, J. M. (1983). Motivational design of instruction. In C. M. Reigeluth (Ed.), Instructional design theories and models (pp. 386-433). Hillsdale, NJ: Erlbaum.Kim, S. (2009). Questioning the stability of foreign language classroom anxiety and motivation across different classroom contexts. Foreign Language Annals, 42(1), 138-157. Komiyama, R. (2009). CAR: A means for motivating students to read. English Teaching Forum, 47(3), 32-37.Kondo-Brown, K. (2001). Bilingual heritage students’ language contact and motivation. In Z.Dörnyei & R. Schmidt (Eds.), Motivation and language acquisition (pp. 433-460).Honolulu, HI: University of Hawai’i Press.Koromilas, K. (2011). Obligation and motivation. Cambridge ESOL Research Notes, 44, 12-20.Lamb, M. (2004). Integrative motivation in a globalizing world. System, 32, 3-19.Lamb, T. (2009). Controlling learning: Learners’ voices and relationships between motivation and learner autonomy. In R. Pemberton, S. Toogood, & A. Barfield (Eds.), Maintaining control: Autonomy and language learning (pp. 67-86). Hong Kong: Hong KongUniversity Press.Lau, K.-l., & Chan, D. W. (2003). Reading strategy use and motivation among Chinese good and poor readers in Hong Kong. Journal of Research in Reading, 26, 177-190.Lepper, M. R. (1983). Extrinsic reward and intrinsic motivation: implications for the classroom.In J. M. Levine & M. C. Wang (eds.), Teacher and student perceptions: implications for learning (pp. 281-317). Hillsdale, NJ; Erlbaum.Li, J. (2009). Motivational force and imagined community in ‘Crazy English.’ In J. Lo Bianco, J.Orton, & Y. Gao (Eds.), China and English: Globalisation and the dilemmas of identity(pp. 211-223). Clevedon, UK: Multilingual Matters.Lopez., F. (2010).Identity and motivation among Hispanic ELLs. Education Policy Analysis Archives, 18(16), 1-29.MacIntyre, P. D. (2002). Motivation, anxiety and emotion in second language acquisition. In P.Robinson (Ed.), Individual differences and instructed language learning (pp. 45-68).Amsterdam, The Netherlands: John Benjamins.MacIntyre, P. D., MacMaster, K., & Baker, S. C. (2001). The convergence of multiple models of motivation for second language learning: Gardner, Pintrich, Kuhl, and McCroskey. In Z.Dörnyei & R. Schmidt (Eds.), Motivation and language acquisition (pp. 461-492).Honolulu, HI: University of Hawai’i Press.Maehr, M. L. & Archer, J. (1987). Motivation and school achievement. In L. G. Katz (ed.), Current topics in early childhood education (pp. 85-107). Norwood, NJ: Ablex.Manderlink, G., & Harackiewicz, J. M. 1984. Proximal versus distal goal setting and intrinsic motivation. Journal of Personality and Social Psychology, 47, 918-928.Markus, H. R., & Kitayama, S. (1991). Culture and self: Implications for cognition, emotion, and motivation. Psychological Review, 98(2), 224-253.Maslow, A. H. (1970). Motivation and personality. New York, NY: Harper & Row.McCombs, B. L. (1984). Processes and skills underlying continued motivation to learn.Educational Psychologist, 19(4), 199-218.McCombs, B. L. (1988). Motivational skills training: combining metacognitive, cognitive, and affective learning strategies. In C. E. Weinstein, E. T. Goetz & P. A. Alexander (Eds.),Learning and study strategies (pp.141-169). New York: Academic Press.McCombs, B. L. (1994). Strategies for assessing and enhancing motivation: keys to promoting self-regulated learning and performance. In H. F. O’Neill & M. Drillings (Eds.),Motivation: theory and research (pp. 49-69). Hillsdale, NJ: Erlbaum.McCombs, B. L., & Whisler, J. S. (1997). The learner-centered classroom and school: Strategies for increasing student motivation and achievement. San Francisco. CA:Jossey-Bass.McCrossan, L. (2011). Progress, motivation and high-level learners. Cambridge ESOL Research Notes, 44, 6-12.Melvin, B. S., & Stout, D. S. (1987). Motivating language learners through authentic materials.In W. Rivers (Ed.), Interactive language teaching (pp. 44–56). New York, NY:Cambridge University Press.Midraj, S., Midraj, J., O’Neil, G., Sellami, A., & El-Temtamy, O. (2007). UAE grade 12 students’ motivation & language learning. In S. Midraj, A. Jendli, & A. Sellami (Eds.), Research in ELT contexts (pp. 47-62). Dubai: TESOL Arabia.Molden, D. C., & Dweck, C. S. (2000). Meaning and motivation. In C. Sansone & J.Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimalmotivation and performance (pp. 131-159). San Diego, CA: Academic Press.Mori, S. (2002). Redefining motivation to read in a foreign language. Reading in a Foreign Language, 14, 91-110.Noels, K. A., Clément, R., & Pelletier, L. G. (1999). Perceptions of teachers’ communicative style and students’ intrinsic and extrinsic motivation.Modern Language Journal,83, 23-34.Noels, K. A., Pelletier, L. G., Clément, R., & Vallerand, R. J. (2000). Why are you learning a second language?: Motivational orientations and self-determination theory. LanguageLearning, 50, 57-85.Oxford, R. L. (Ed.). (1996). Language learning motivation: pathways to the new century.Hono lulu: Second Language Teaching & Curriculum Center, University of Hawai’i.Oxford, R., & Shearin, J. (1994). Language learning motivation: Expanding the theoretical framework. Modern Language Journal, 78, 12-28.Papadimitriou, A. D. (2011). The impact of an extensive reading programme on vocabulary development and motivation. Cambridge ESOL Research Notes, 44, 39-47.Paris, S. C., & Turner, J. C. (1994). Situated motivation. In P. R. Pintrich, D. R. Brown, & C. E.Weinstein (Eds.), Student motivation, cognition, and learning (pp. 213-237). Hillsdale,NJ: Erlbaum.Peacock, M. (1997). The effect of authentic materials on the motivation of EFL learners. ELT Journal, 51(2), 144-156.Pedraza, P., & Ayala, J. (1996). Motivation as an emergent issue in an after-school program in El Barrio. In L. Schauble & R. Glaser (Eds.), Innovations in learning (pp. 75-91). Mahwah, NJ: Erlbaum.Pintrich, P. R. (1999). The role of motivation in promoting and sustaining self-regulated learning.International Journal of Educational Research, 31(6), 459-470.Pintrich, P. R., & De Groot, E. V. (1990). Motivational and self-regulated learning components of classroom academic performance. Journal of educational psychology, 82(1), 33.Pintrich, P. R., & Schunk, D. H. (1996). Motivation in education: Theory, research and applications. Englewood Cliffs: NJ: Prentice-Hall.Ramage, K. (1991). Motivational factors and persistence in second language learning. Language Learning, 40(2), 189-219.Rueda, R., & Moll, L. (1994). A sociocultural perspective on motivation. In H. F. O’Neill & M.Drillings (Eds.), Motivation: theory and research (pp. 117-137). Hillsdale, NJ: Erlbaum.Ryan, R. M., & Deci, E. L. (2000). Intrinsic and extrinsic motivations: Classic definitions and new directions. Contemporary Educational Psychology,25, 54-67.Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55, 68-78. Schmidt, R., Boraie, D., & Kassabgy, O. (1996). Foreign language motivation: Internal structure and external connections. In R. Oxford (Ed.), Language learning motivation:Pathways to the new century (pp. 9-70). Honolulu: Second Language Teaching &Curriculum Center, University of Hawaii Press.Schutz, P. A. (1994). Goals as the transaction point between motivation and cognition. In P. R.Pintrich, D. R. Brown, & C. E. Weinstein (Eds.), Student motivation, cognition, andlearning (pp. 135-156). Hillsdale, NJ: Erlbaum.Scott, K. (2006). Gender differences in motivation to learn French. The Canadian Modern Language Review, 62, 401-422.Syed, Z. (2001). Notions of self in foreign language learning: A qualitative analysis. In Z.Dörnyei & R. Schmidt (Eds.), Motivation and second language acquisition (pp. 127-147).Honolulu, HI: University of Hawaii Press.Schmidt, R., Boraie, D., & Kassabgy, O. (1996). Foreign language motivation: Internal structure and external connections. In R. Oxford (Ed.), Language learning motivation: Pathways to the new century (pp. 9-70). Honolulu, HI: Second Language Teaching & Curriculum Center, University of Hawaii Press.Scott, K. (2006). Gender differences in motivation to learn French. The Canadian Modern Language Review, 62, 401-422.Sisamakis, M. (2010). The motivational potential of the European language portfolio. In B.O’Rourke & L. Carson (Eds.), Language learner autonomy: Policy, curriculum,classroom (pp. 351-371). Oxford, UK: Peter Lang.Smith, K. (2006). Motivating students through assessment. In R. Wilkinson, V. Zegers, & C. van Leeuwen (Eds.), Bridging the assessment gap in English-medium higher education (pp.109-121). Nijmegen, The Netherlands: AKS –Verlag.St. John, J. (2007). Motivation: The teachers’ perspective. In S. Midraj, A. Jendli, & A. Sellami (Eds.), Research in ELT contexts (pp. 63-84). Dubai: TESOL Arabia.Syed, Z. (2001). Notions of self in foreign language learning: a qualitative analysis. In Z.Dörnyei & R. Schmidt, (Eds.), Motivation and second language acquisition (pp. 127-148).Honolulu: National Foreign Language Resource Center/ University of Hawai'iPress.Ushioda, E. (2003). Motivation as a socially mediated process. In D. Little, J. Ridley, & E.Ushioda (Eds.), Learner autonomy in the language classroom (pp. 90-102). Dublin,Ireland: Authentik.Ushioda, E. (2008). Motivation and good language learners. In C. Griffiths (Ed.), Lessons from good language learners (pp. 19-34). Cambridge, UK: Cambridge University Press. Ushioda, E. (2009). A person-in-context relational view of emergent motivation, self and identity.In Z. Dörnyei & E. Ushioda (Eds.), Motivation, language identity and the L2 self (pp.215-228). Tonawanda, NY: Multilingual Matters.Ushioda, E. (2012). Motivation. In A. Burns & J. C. Richards (Eds.), The Cambridge guide to pedagogy and practice in language teahcing (pp. 77-85). Cambridge, UK: CambridgeUniversity Press.Verhoeven, L., & Snow, C. E. (Eds.). (2001). Literacy and motivation: Reading engagement in individuals and groups. Mahwah, NJ: Lawrence Erlbaum.Wachob, P. (2006). Methods and materials for motivation and learner autonomy. Reflections on English Language Teaching, 5(1), 93-122.Walker, C. J., & Quinn, J. W. (1996). Fostering instructional vitality and motivation. In R. J.Menges and associates, Teaching on solid ground (pp. 315-336). San Francisco, SF:Jossey-Bass.Warden, C. A., & Lin, H. J. (2000). Existence of integrative motivation in an Asian EFL setting.Foreign Language Annals, 33, 535-547.Weiner, B. (1984). Principles for a theory of student motivation and their application within an attributional framework. In R. Ames & C. Ames (Eds.), Research on motivation ineducation (pp. 15-38). New York: Academic Press.Weiner, B. (1992). Human motivation: Metaphors, theories and research. Newbury Park, CA: Sage.Weiner, B. (1994). Integrating social and personal theories of achievement motivation. Review of Educational Research, 64, 557-573.Williams, M., Burden, R., & Lanvers, U. (2002). French is the language of love and stuff: Student perceptions of issues related to motivation in learning a foreign language.British Educational Research Journal, 28, 503–528.Wlodkowski, R. J. (1985). Enhancing adult motivation to learn. San Francisco, CA: Jossey-Bass. Wu, X. (2003). Intrinsic motivation and young language learners: The impact of the classroom environment. System, 31, 501-517.Xu, H., & Jiang, X. (2004). Achievement motivation, attributional beliefs, and EFL learning strategy use in China. Asian Journal of English Language Teaching, 14, 65-87.Yung, K. W. H. (2103). Bridging the gap: Motivation in year one EAP classrooms. Hong Kong Journal of Applied Linguistics, 14(2), 83-95.。

XANES_intro

basics Selection rules (LS coupling)
Text
source /Pubs/AtSpec/node17.html
radial wavefunctions
• electric dipole selection rules require p ﬁnal
The literature abounds with empirical correlations between edge shifts and formal charge state, and the relationships between pre-edge transitions and symmetry The purpose of this talk to try to dig below the surface of these correlations without getting buried in mathematics. I recommend Simon Bare’s talk on XANES (2005 APS XAFS school) for a good overview of the applications and very interesting ﬁgures, and Bruce Ravel’s APS XAFS school) talk on applying the FEFFx program to modeling data.
(consequence of Pauli exclusion principle)
• ﬁlling of orbital suppresses white line
Core Hole Lifetime
The core hole (vacancy in the initial state following x-ray absorption) is unstable. It decays in a short time ~ 1 femtosecond, typically by emitting a ﬂuorescence photon or auger electron. By the uncertainty principle, the energy width of a state is inversely proportional to its lifetime. Higher atomic number elements have shorter lifetimes -> greater broadening. This gets to be a limitation for high Z edges although it can be deconvoluted out in favorable cases. Low atomic number (~Z=8) spectra “NEXAFS” can be interpreted in great detail because of sharp spectral lines X-ray Inelastic Scattering can be used to suppress core hole lifetime broadening and explore XANES in detail

ledata

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The high-speed performance and library design reuse features significantly enhance a workgroup environment.In addition, the easy-to-use interface allows designers to increase productivity without the steep learning curve so you can concentrate on your design, not on the tool.Support for industry standard file formats, such as GDSII, CIF , SPICE, and EDIF allows convenient interfacing with other tools. Fully customizable shortcut key mapping and layer/background display equips designers with a comfortable, personalized design environment.VERIFICATIONDESIGN RULE CHECKING (DRC)L-Edit/DRC™ is a user-configurable design rule checker that verifies layout against thestandard design rule sets provided by most foundries. DRC rules are easily setup throughthe graphical interface, or can be conveniently imported from a Dracula® script. Use DRCto verify a full chip or just a specific design region.Boolean and Select operations are supported, and 90˚ and 45˚ geometry checks are alsoincluded. Errors are reported to the screen, where they are easily browsed using theDRC Error Navigator to browse by rule type.•Width, spacing, surround, overlap, extension, not exists, and density rules•Boolean operations AND, OR, NOT, and Grow/Shrink supported•Inside, outside, hole, cut, touch, enclose, overlap and vertex select operationssupported for layer derivations•Flagging for off-grid and self-intersecting polygons•Dracula® rule set import translator•Import Calibre® DRC results for browsing in L-EditPARAMETERIZED CELLST-CELLST-Cells provide the ability to create parameterized cells based on user-written C-language code.T-Cells extend traditional geometry cells to include the flexibility and automation of L-Edit’s user-programmable interface (UPI).With T-Cells, a designer can create versatile source cells that consist of user-defined input parametersand layout-generating code, as well as optional fixed geometry. 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0411523v1 Twisted representations of vertex operator superalgebras

a r X i v :m a t h /0411523v 1 [m a t h .Q A ] 23 N o v 2004Twisted representations of vertex operatorsuperalgebrasChongying Dong 1and Zhongping ZhaoDepartment of Mathematics,University of California,Santa Cruz,CA 95064AbstractThis paper gives an analogue of A g (V )theory for a vertex operator superalgebra V and an automorphism g of ﬁnite order.The relation between the g -twisted V -modules and A g (V )-modules is established.It is proved that if V is g -rational,then A g (V )is ﬁnite dimensional semisimple associative algebra and there are only ﬁnitely many irreducible g -twisted V -modules.1IntroductionThe twisted sectors or twisted modules are basic ingredients in orbifold conformal ﬁeld theory (cf.[FLM1],[FLM2],[FLM3],[Le1],[Le2],[DHVW],[DVVV],[DL2],[DLM2]).The notion of twisted module [FFR],[D]is derived from the properties of twisted vertex operators for ﬁnite automorphisms of even lattice vertex operator algebras constructed in [Le1],[Le2]and [FLM2],also see [DL2].In this paper we study the twisted modules for an arbitrary vertex operator superalgebra following [Z],[KW]and [DLM2].An associative algebra A (V )was introduced in [Z]for every vertex operator algebra V to study the representation theory for vertex operator algebra.The main idea is to reduce the study of representation theory for a vertex operator algebra to the study of represen-tation theory for an associative algebra.This approach has been very successful and the irreducible modules for many well-known vertex operator algebras have been classiﬁed by using the associative algebras.This theory has been extended to the vertex operator superalgebras in [KW]and has been further generalized to the twisted representations for a vertex operator algebra in [DLM2].This paper is a “super analogue”of [DLM2].We construct an associative algebra A g (V )for any vertex operator superalgebra V together with an automorphism g of ﬁnite order.Then the vacuum space of any admissible g -twisted V -module becomes a module for A g (V ).On the other hand one can construct a ‘universal’admissible g -twisted V -module from any A g (V )-module.This leads to a one to one correspondence between the set of inequivalent admissible g -twisted V -modules and the set of simple A g (V )-modules.As in the case of vertex operator algebra,if V is g -rational then A g (V )is a ﬁnite dimensional semisimple associative algebra.The ideas of this paper and other related papers are very natural and go back to the theory of highest weight modules for Kac-Moody Lie algebras and other Lie algebras with triangular decompositions.In the classical highest weight module theory,the highestweight or highest weight vector determines the highest weight module structure to some extend(diﬀerent highest weight modules can have the same highest weight).The role of the vacuum space for an admissible twisted module is similar to the role of the highest weight space in a highest weight module.So from this point of view,the A g(V)theory is a natural extension of highest weight module theory in the representation theory of vertex operator superalgebras.A vertex operator superalgebra has a canonical automorphismσof order2arising from the structure of superspace.Theσ-twisted modules which are called the Ramond sector in the literature play very important roles in the study of geometry.Important topological invariants such as elliptic genus and certain Witten genus can be understood as graded trace functions on the Ramond sectors constructed from the manifolds.It is expected that the theory developed in this paper will have applications in geometry and physics.Since the setting and most results in this paper are similar to those in[DLM2]we only provide the arguments which are either new or need a lot of modiﬁcations.We refer the reader to[DLM2]for details.The organization of this paper is similar to that of[DLM2].We review the deﬁnition of vertex operator superalgebra and deﬁne various notions of g-twisted V-modules in section2.In section3,we introduce the algebra A g(V)for VOSA V.Section4is devoted to the study of Lie superalgebra V[g]which is kind of twisted aﬃnization of V.A weak g-twisted V-module is naturally a V[g]-module.In section5,we construct the functorΩwhich sends a weak g-twisted V-module to an A g(V)-module.We construct another functor L from the category of A g(V)-modules to the category of admissible g-twisted V-modules in Section6.That is,for any A g(V)-module U we can construct a kind of“generalized Verma module”¯M(U)which is the universal admissible g-twisted V-module generated by U.It is proved that there is a1-1correspondence between the irreducible objects in these two categories.Moreover if V is g-rational,then A g(V)is a ﬁnite dimensional semisimple associative algebra.We discuss some examples of vertex operator superalgebras constructed from the free fermions and their twisted modules in Section7.2Vertex Operator superalgebra and twisted mod-ulesWe review the deﬁnition of vertex operator superalgebra(cf.[B],[FLM3],[DL1])and various notions of twisted modules in this section(cf.[D],[DLM2],[FFR],[FLM3],[Z]).Recall that a super vector space is a Z2-graded vector space V=V¯0⊕V¯1.The elements in V¯0(resp.V¯1)are called even(resp.odd).Let˜v be0if v∈V¯0,and1if v∈V¯1.Deﬁnition2.1.A vertex operator superalgebra is a12Z+V n=V¯0⊕V¯1.(2.1)with V¯0= n∈Z V n and V¯1= n∈112(m3−m)δm+n,0c;(2.6)dz0 Y(u,z1)Y(v,z2)−(−1)˜u˜v z−10δ z2−z1z2 Y(Y(u,z0)v,z2).(2.9) whereδ(z)= n∈Z z n and(z i−z j)n is expanded as a formal power series in z j.Throughout the paper,z0,z1,z2,etc.are independent commuting formal variables.Such a vertex operator superalgebra may be denoted by V=(V,Y,1,ω).In the case V¯1=0,this is exactly the deﬁnition of vertex operator algebra given in[FLM3].Deﬁnition2.2.Let V be a vertex operator superalgebra.An automorphism g of V is a linear automorphism of V preservingωsuch that the actions of g and Y(v,z)on V are compatible in the sense thatgY(v,z)g−1=Y(gv,z)for v∈V.Note that any automorphism of V commutes with L(0)and preserves each homoge-neous space V n.As a result,any automorphism preserves V¯0and V¯1.Let Aut(V)be the group of automorphisms of V.There is a special automorphism σ∈Aut(V)such thatσ|V¯0=1andσ|V¯1=−1.It is clear thatσis a central element of Aut(V).Fix g∈Aut(V)of order T0.Let o(gσ)=T.Denote the decompositions of V into eigenspaces with respect to the actions of gσand g as followsV=⊕r∈Z/T Z V r∗(2.10)V=⊕r∈Z/T0ZV r(2.11) where V r∗={v∈V|gσv=e2πir/T v}and V r={v∈V|gv=e2πir/T0v}Deﬁnition2.3.A weak g-twisted V-module M is a vector space equipped with a linear mapV→(End M)[[z1/T0,z−1/T0]v→Y M(v,z)= n∈1T0+Zu n z−n−1;(2.12)u l w=0for l>>0;(2.13)Y M(1,z)=Id M;(2.14) z−10δ z1−z2−z0 Y M(v,z2)Y M(u,z1)=z−12 z1−z0z2 Y M(Y(u,z0)v,z2).(2.15)Following the arguments in[DL1]one can prove that the twisted Jacobi identity is equivalent to the following associativity formula(z0+z2)k+r T0Y M(Y(u,z0)v,z2)w.(2.16) where w∈M and k∈Z+s.t z k+rz2 −r/T0δz1−z0Lemma2.4.The associativity formula(2.16)is equivalent to the following: (z0+z2)m+s T Y M(Y(u,z0)v,z2)wfor u∈V s∗and some m∈1T Y M(u,z)w involves only nonnegative integral powers of z.Proof:Let u∈V r.It is enough to prove that wt u+sT0are congruent modulo Z.It is easy to see that s≡T2˜u+2r modulo Z if T0is odd.Thus wt u+s2˜u+r2˜u and wt u are congruentmodulo Z,the result follows immediately.Equating the coeﬃcients of z−m−11z−n−12in(2.17)yields[u m,v n]=∞i=0 m i (u i v)m+n−i.(2.18)We may also deduce from(2.12)-(2.15)the usual Virasoro algebra axioms,namely that if Y M(ω,z)= n∈Z L(n)z−n−2then[L(m),L(n)]=(m−n)L(m+n)+1dzY M(v,z)=Y M(L(−1)v,z)(2.20) (cf.[DLM1]).The homomorphism and isomorphism of weak twisted modules are deﬁned in an ob-vious way.Deﬁnition2.5.An admissible g-twisted V-module is a weak g-twisted V-module M which carries a1T Z+M(n)(2.21)satisfyingv m M(n)⊆M(n+wt v−m−1)(2.22) for homogeneous v∈V.Deﬁnition2.6.An ordinary g-twisted V-module is a weak g-twisted V-moduleM= λ∈C Mλ(2.23) such that dim Mλisﬁnite and forﬁxedλ,M nThe admissible g-twisted V-modules form a subcategory of the weak g-twisted V-modules.It is easy to prove that an ordinary g-twisted V-module is admissible.Shifting the grading of an admissible g-twisted module gives an isomorphic admissible g-twisted V-module.A simple object in this category is an admissible g-twisted V-module M such that0and M are the only graded submodules.We say that V is g-rational if every admissible g-twisted V-module is completely reducible,i.e.,a direct sum of simple admissible g-twisted modules.V is called rational if V is1-rational.V is called holomorphic if V is rational and V is the only irreducible V-module up to isomorphism.If M=⊕n∈1T Z+M(n)∗(2.24)where M(n)∗=Hom C(M(n),C).The vertex operator Y M′(a,z)is deﬁned for a∈V via Y M′(a,z)f,u = f,Y M(e zL(1)(−z−2)L(0)a,z−1)u (2.25) where · denotes the natural paring between M′and M.Then we have the following [FHL]:Lemma2.7.(M′,Y M′)is an admissible g−1-twisted V-module.Lemma2.7is needed in the proof of several results in Section6although we do not intend to give these proofs(cf.[DLM2]).3The associative algebra A g(V)Let r be an integer between0and T−1(or T0−1).We will also use r to denote its residue class modulo T or T0.For homogeneous u∈V r∗,we setδr=1if r=0andδr=0 if r=0.Let v∈V we deﬁneu◦g v=Res z (1+z)wt u−1+δr+rz1+δrY(u,z)v(3.1)where(1+z)αforα∈C is to be expanded in nonnegative integer powers of z.Let O g(V) be the linear span of all u◦g v and deﬁne the linear space A g(V)to be the quotient V/O g(V).We will use A(V),O(V),u◦v,when g=1.The A(V)was constructed in[KW] and if V is a vertex operator,A g(V)was constructed in[DLM2].Lemma3.1.If r=0then V r∗⊆O g(V).Proof:The proof is the same as that of Lemma2.1in[DLM2].Let I=O g(V)∩V0∗.Then A g(V)≃V0∗/I(as linear spaces).Since O(V0∗)⊂I, A g(V)is a quotient of A(V0∗).We now deﬁne a product ∗g on V which will induce an associative product in A g (V ).Let r,u and v be as above and setu ∗gv =Res z (Y (u,z )(1+z )wt uT+nzY (v,z )u ∈O (V 0∗)and(iii)u ∗v −(−1)˜u ˜v v ∗u −Res z (1+z )wt u −1Y (u,z )v ∈O (V 0∗).Proof:See the proofs of Lemmas 2.1.2and 2.1.3of[Z]bynoting thatY (u,z )v ≡(−1)˜u ˜v (1+z )−wtu −wtv Y (v,−zz Y (c,z )u(3.4)andu∗c≡Res z(1+z)wt c−1z0 Y(c,z1)Y(a,z2)b−(−1)˜c˜a z−10δ z2−z1z2 Y(Y(c,z0)a,z2)b.(3.6) Forε=0or1,(3.6)implies:xε=Res z1(1+z1)wt c−εTz1Y(c,z1)(1+z2)wt a−1+δr+rz1+δr2Y(a,z2)b=(−1)˜a˜c Res z1Res z2(1+z1)wt c−εTz1(1+z2)wt a−1+δr+rz1+δr2z−12δ z1−z0z1(1+z2)wt a−1+δr+rz1+δr2Y(a,z2)Y(c,z1)b+Res z2Res z(1+z2+z0)wt c−εTTz1Y(c,z1)b+∞i,j=0(−1)j wt c−εi Res z2(1+z2)wt a−1+δr+r z j+2+δr2Y(c i+j a,z2)b=(−1)˜c˜a Res z2(1+z2)wt a−1+δr+rz1+δr2Y(a,z2)Res z1(1+z1)wt c−εT+j+1−εNext we prove that∗g is associative.We need to verify that(a∗b)∗c−a∗(b∗c)∈O g(V0∗)for a,b,c∈V0∗.A straightforward computation using the twisted Jacobi identity gives(a∗b)∗c=wt a i=0(a i−1b)∗c=wt ai=0 wt a i Res w(Y(a i−1b,w)(1+w)wt(a i−1b)wc)=Res w Res z−w(Y(Y(a,z−w)b,w)(1+z)wt a(1+w)wt bw(z−w)c)−(−1)˜a˜b Res w Res z(Y(b,w)Y(a,z)(1+z)wt a(1+w)wt bwc)−(−1)˜a˜b∞i=0Res w Res z(Y(b,w)Y(a,z)(−1)i+1z i w−i−1(1+z)wt a(1+w)wt bzwc)mod O g(V0∗)≡a∗(b∗c)mod O g(V0∗)Thus A g(V)≃V0∗T0,t−1dt f(t) g(t).(4.1) (see[B]).Then the tensor productL(V)=C[t1T0]⊗V.(4.2)is a vertex superalgebra with vertex operatorY (f (t )⊗v,z )(g (t )⊗u )=f (t +z )g (t )⊗Y (v,z )u.(4.3)The L (−1)operator of L (V )is given by D =dT 0)(t m ⊗ga ).(4.4)Let L (V,g )be the g -invariants which is a vertex sub-superalgebra of L (V ).Clearly,L (V,g )=⊕T 0−1r =0tr/T 0C [t,t −1]⊗V r .(4.5)Following [B],we know thatV [g ]=L (V,g )/D L (V,g )(4.6)is a Lie superalgebra with bracket[u +D L (V,g ),v +D L (V,g )]=u 0v +D L (V,g ).(4.7)For short let a (q )be the image of t q ⊗a ∈L (V,g )in V [g ].Then we have Lemma 4.1.Let a ∈V r ,v ∈V s and m,n ∈Z .Then(i)[ω(0),a (m +r T 0a (m −1+rT 0),b (n +s T 0ia ib (m +n +r +sTZ -graded.Since D increases degree by 1,D L (V,g )is a graded subspace ofL (V,g )and V [g ]is naturally 1TZV [g ]n .By Lemma 4.1,V [g ]is a1TZV [g ]±n .Lemma 4.2.V [g ]0is spanned by elements of the form a (wt a −1)for homogeneous a ∈V 0∗.Proof:Let a∈V.Then the degree wt a−n−1of a(n)is0if and only if a∈V0¯andn=wt a−1or a∈V T0/2¯1and n=wt a−1.The bracket of V[g]0is given by[a(wt a−1),b(wt b−1)]=∞j=0 wt a−1j a j b(wt(a j b)−1).(4.10)Set o(a)=a(wt a−1)for homogeneous a∈V0∗and extend linearly to all a∈V0∗. This gives a linear mapV0∗→V[g]0,a→o(a).(4.11) As the kernel of the map is(L(−1)+L(0))V0∗,we obtain an isomorphism of Lie super-algebras V0∗/(L(−1)+L(0))V0∗∼=V[g]0.The bracket on the quotient of V0∗is given by[a,b]= j≥0 wt a−1j a j b.Lemma4.3.Let A g(V)Lie be the Lie superalgebra of the associative algebra A g(V)intro-duced in section3such that[u,v]=u∗g v−(−1)˜u˜v v∗g u.Then the map o(a)→a+O g(V) is an onto Lie superalgebra homomorphism from V[g]0to A g(V)Lie.Proof:Recall that I=O g(V)∩V0∗.So we have a surjective linear mapV[g]0∼=V0∗/(L(−1)+L(0))V0∗→V0∗/I≃A g(V),o(a)→a+(L(−1)+L(0))V0∗→a+I.(4.12) The Lie homomorphism follows from[o(a),o(b)]=∞j=0 wt a−1j o(a j b).and[a+O g(V),b+O g(V)]≡a∗g b−(−1)˜a˜b b∗g a≡∞j=0 wt a−1j a j b≡Res z(1+z)wt a−1Y(a,z)b mod O g(V0∗)≡∞i=0 wt a−1i a i b mod O g(V0∗).5The functorΩThe main purpose in this section is to construct a covariant functorΩfrom the category of weak g-twisted V-modules to the category of A g(V)-modules(cf.Theorem5.1).Let M be a weak g-twisted V-module.We deﬁne the space of“lowest weight vectors”to beΩ(M)={w∈M|u wt u+n w=0,u∈V,n≥0}.The main result in this section says thatΩ(M)is an A g(V)-module.Moreover if f:M→N is a morphism in weak g-twisted V-modules,the restrictionΩ(f)of f toΩ(M)is an A g(V)-module morphism.Note that if M is a weak g-twisted V-module then M becomes a V[g]-module such that a(m)acts as a m.Moreover,M is an admissible g-twisted V-module if and only if M is a1TY(u,z)v.z2The argument in the Proof of Theorem2.1.2in[Z]with suitable modiﬁcation giveso(u∗v)=o(u)o(v).Note that o(L(−1)u+L(0)u)=0and(L(−1)u+L(0)u)∗v=u◦v.We immediately have o(u◦v)=0onΩ(M).Ifa=Res z (1+z)wt c−1+rzY(u,z)v,we can use Lemma2.4.Since z wt u−1+rT Y M(u,z0+z2)Y M(v,z2)w=(z2+z0)wt u−1+rT2to(5.1)yields0=Res z0Res z2z−10zwt v−rT Y M(Y(u,z0)v,z2)w=∞i=0 wt u−1+rTi o(u i−1v)w=o Res z(1+z)wt u−1+r z Y M(u,z)v w=o(a)w(5.2) as required.If M is a nonzero admissible g-twisted V-modules we may and do assume that M(0) is nonzero with suitable degree shift.With these conventions we haveProposition5.2.Let M be a simple admissible g-twisted V-module.Then the following hold(i)Ω(M)=M(0).(ii)Ω(M)is a simple A g(V)-module.Proof:The proof is the same as in[DLM2].6Generalized Verma modules and the functor LIn this section we focus on how to construct admissible g-twisted V-modules from a given A g(V)-module U.We use the same trick which was used in[DLM2]to do this.We will deﬁne two g-twisted admissible V-modules¯M(U)and L(U).The¯M(U)is the universal admissible g-twisted V-module such that¯M(U)(0)=U and L(U)is smallest admissible g-twisted V-module whose L(U)(0)=U.Just as in the classical highest weight module theory,L(U)is the unique irreducible quotient of¯M(U)if U is simple.We start with an A g(V)-module U.Then U is automatically a module for A g(V)Lie.By Lemma4.3U is lifted to a module for the Lie superalgebra V[g]0.Let V[g]−act trivially on U and extend U to a P=V[g]−⊕V[g]0-module.Consider the induced moduleM(U)=Ind V[g]P(U)=U(V[g])⊗U(P)U(6.1)T0Zv(m)z−m−1(6.2)Then Y M(U)(v,z)satisﬁes condition(2.12)-(2.14).By Lemma4.1(ii),the identity(2.18) holds.But this is not good enough to establish the twisted Jacobi identity for the action (6.2)on M(U).Let W be the subspace of M(U)spanned linearly by the coeﬃcients of(z0+z2)wt a−1+δr+r T Y(Y(a,z0)b,z2)u(6.3) for any homogeneous a∈V r∗,b∈V,u∈U.We set¯M(U)=M(U)/U(V[g])W.(6.4) Proposition6.1.Let M be a V[g]-module such that there is a subspace U of M satisfying the following conditions:(i)M=U(V[g])U;(ii)For any a∈V r∗and u∈U there is k∈wt a+Z+such that(z0+z2)k+r T Y(Y(a,z0)b,z2)u(6.5) for any b∈V.Then M is a weak V-module.Proof:We only need to prove the twisted Jacobi identity,which is equivalent to com-mutator relation(2.17)and the associativity(2.16).But the commutator formula is built in already as M is a V[g]-module.By Lemma2.4,the assumption(ii)can be reformulated as follows:(ii’)For any a∈V r and u∈U there is k∈Z+such that(z0+z2)k+r T0Y(Y(a,z0)b,z2)u(6.6) Since M is a V[g]-module generated by U it is enough to prove that if u satisﬁes(ii’) then c n u also satisﬁes(ii’)for c∈V and n∈1T0Y(c i a,z0+z2)Y(b,z2)u=(z2+z0)k2+r+sT0Y(a,z0+z2)Y(c i b,z2)u=(z2+z0)k2+r+sT0+n−k1>k2+r+s(z 0+z 2)k +rT 0c n Y (a,z 0+z 2)Y (b,z 2)u −(−1)˜a ˜c (−1)˜b ˜c∞ i =0n i (z 0+z 2)k +r T 0Y (a,z 0+z 2)Y (c i b,z 2)u=(−1)˜a ˜c (−1)˜b ˜c (z 0+z 2)k +rT 0+n −iY (Y (c i a,z 0)b,z 2)u−(−1)˜b ˜c∞ i =0n iz n −i2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c ∞ i =0n i (z 2+z 0)k +r T 0Y (c i Y (a,z 0)b,z 2)u+(−1)˜a ˜c (−1)˜b ˜c ∞ i =0∞ j =0n j j iz n −i2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c ∞ i =0n i (z 2+z 0)k +r T 0Y (c i Y (a,z 0)b,z 2)u+(−1)˜a ˜c (−1)˜b ˜c ∞ j =0∞ i =jn j n −ji −jz n −i2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c∞ i =0n i z n −i 2(z 2+z 0)k +rT 0c n Y (Y (a,z 0)b,z 2)u−(−1)˜a ˜c (−1)˜b ˜c (z 2+z 0)k +rT 0Y (Y (a,z 0)b,z 2)c n u,=(z 2+z 0)k +rThe proof is complete.Applying Proposition6.1to¯M(U)gives the following main result of this section. Theorem6.2.¯M(U)is an admissible g-twisted V-module with¯M(U)(0)=U and with the following universal property:for any weak g-twisted V-module M and any A g(V)-morphismφ:U→Ω(M),there is a unique morphism¯φ:¯M(U)→M of weak g-twisted V-modules which extendsφ.As in[DLM2]we also haveTheorem6.3.M(U)has a unique maximal graded V[g]-submodule J with the property that J∩U=0.Then L(U)=M(U)/J is an admissible g-twisted V-module satisfying Ω(L(U))∼=U.L deﬁnes a functor from the category of A g(V)-modules to the category of admissible g-twisted V-modules such thatΩ◦L is naturally equivalent to the identity.We have a pair of functorsΩ,L between the A g(V)-module category and admissible g-twisted V-module category.AlthoughΩ◦L is equivalent to the identity,L◦Ωis not equivalent to the identity in general.The following result is an immediate consequence of Theorem6.3.Lemma6.4.Suppose that U is a simple A g(V)-module.Then L(U)is a simple admissible g-twisted V-module.Using Lemma6.4,Proposition5.2(ii),Theorems6.2and6.3gives:Theorem6.5.L andΩare equivalent when restricted to the full subcategories of com-pletely reducible A g(V)-modules and completely reducible admissible g-twisted V-modules respectively.In particular,L andΩinduces mutually inverse bijections on the isomor-phism classes of simple objects in the category of A g(V)-modules and admissible g-twisted V-modules respectively.We now apply the obtained results to g-rational vertex operator superalgebras to obtain:Theorem6.6.Suppose that V is a g-rational vertex operator superalgebra.Then the following hold:(a)A g(V)is aﬁnite-dimensional,semi-simple associative algebra(possibly0).(b)V has onlyﬁnitely many isomorphism classes of simple admissible g-twisted mod-ules.(c)Every simple admissible g-twisted V-module is an ordinary g-twisted V-module.(d)V is g−1-rational.(e)The functors L,Ωare mutually inverse categorical equivalences between the cate-gory of A g(V)-modules and the category of admissible g-twisted V-modules.(f)The functors L,Ωinduce mutually inverse categorical equivalences between the category ofﬁnite-dimensional A g(V)-modules and the category of ordinary g-twisted V-modules.The proof is the same as that of Theorem8.1in[DLM2].7ExamplesIn this section we discuss the well known vertex operator superalgebras constructed from the free fermions and their twisted modules.In particular we compute the algebra A g (V )and classify the irreducible twisted modules using A g (V ).The classiﬁcation results have been obtained previously in [Li2]with a diﬀerent approach.Let H = li =1C a i be a complex vector space equipped with a nondegenerate symmet-ric bilinear form (,)such that {a i |i =1,2,...l }form an orthonormal basis.Let A (H,Z +12}subject to the relation [a (n ),b (m )]+=(a,b )δm +n,0.Let A +(H,Z +12,n >0},andmake C a 1-dimensional A +(H,Z +12)=A (H,Z +12)C∼=Λ[a i(−n )|n >0,n ∈Z +1∂a i (−n )if n is positive and by multiplication by a i (n )if nis negative.The V (H,Z +12Zso thatV (H,Z +12Zwe deﬁne a normal ordering:b 1(n 1)···b k (n k ):=(−1)|σ|b i 1(n i 1)···b i k (n i k )such that n i 1≤···≤n i k where σis the permutation of {1,...,k }by sending j to i j .For a ∈H set Y (a (−1/2),z )=n ∈12)···b k (−n k −12)where n i arenonnegative integers.We setY (v,z )=:(∂n 1b 1(z ))···(∂n k b k (z )):where ∂n =1dz)n .Then we have a linear map:V (H,Z +12))[[z,z −1]]v→Y (v,z )=n ∈Z v n z −n −1(v n ∈End V (H,Z +12li =1a i (−32).The following result is well known (cf.[FFR],[KW]and [Li1]).Theorem7.1.(V(H,Z+12)for i=1,...,l.We have already mentioned in Section2that any vertex operator superalgebra has a canonical automorphismσsuch thatσ=1on V¯0andσ=−1on V¯1.Note thatV(H,Z+12)¯1.We next discuss theσ-twisted V(H,Z+1∂b i(−n)∗if n is nonnegative and multiplication by b i(n)if n is nega-tive.Similarly,b i(n)∗acts as∂2)-module such thatY V(H,Z)(u(−12)is isomorphic to thematrix algebra M2k×2k(C)and V(H,Z)is the unique irreducibleσ-twisted V(H,Z+12)is isomorphic to thematrix algebra M2k×2k(C).Since g=σ,the decomposition(2.10)becomes V=V0∗.By lemma3.2(i),Res z (1+z)1z2+ma i(z)v= s≥0c s a i(−m+s−3lies in O σ(V (H,Z +12s.This implies that a i (−m −32+s )vmod O σ(V (H,Z +12))is spanned by b 1(−1/2)s 1···b k (−1/2)s k b ∗1(−1/2)t 1···b ∗k (−1/2)t kwith s i ,t i =0,1.As a result,dim A σ(V (H,Z +12)-module.By Theorem 5.1,Ω(V (H,Z ))is a simple A σ(V (H,Z +12)≥dim Ω(V (H,Z ))=22k .This forces dim A σ(V (H,Z +12))∼=M 2k ×2k (C ).We now deal with the case dim H =2k +1for some nonnegative integer k.Then H can be decomposed into:H =k i =1C b i +k i =1C b ∗i +C ewith (b i ,b j )=(b ∗i ,b ∗j )=0,(b i ,b ∗j )=δi,j ,(e,b i )=(e,b ∗i )=0,(e,e )=2.Let A (H,Z )be the associative algebra generated same as above,and A (H,Z )+be the subalgebra generated by {b i (n ),b ∗i (m ),e (n )|m,n ∈Z ,m >0,n ≥0,i =1,···,k }and make C a 1-dimensional A (H,Z )+-module so that b i (n )1=0for n ≥0and b ∗i (m )1=e (m )1=0for m >0,i =1,···,k.SetV (H,Z )=A (H,Z )⊗A (H,Z )+C∼=Λ[b i (−n ),b ∗i (−m ),e (−m )|n,m ∈Z ,n >0,m ≥0]and letW (H,Z )=Λ[b i (−n ),b ∗i (−m ),e (−n )|n,m ∈Z ,n >0,m ≥0]=W (H,Z )even⊕W (H,Z )odd be the decomposition into the even and old parity subspaces.Also deﬁneV ±(H,Z )=(1±e (0))W (H,Z )even ⊕(1∓e (0))W (H,Z )odd .ThenV (H,Z )=V +(H,Z )⊕V −(H,Z )and V ±(H,Z )are irreducible A (H,Z )-modules.The actions of b i (n ),b ∗i (n )are the same as before.The e (n )acts as 2∂2),z )=u (z )=n ∈Zu (n )z −n −1/2for u ∈H.Proposition7.3.If dim H=2k+1is odd,then Aσ(V(H,Z+12)has exactly two irreducibleσ-twisted modulesV±(H,Z)up to isomorphism.Proof:The proof is similar to that of Proposition7.2.Note that the automorphismσof V(H,Z+12)as follows:For any a1(−n1)···a s(−n s)∈V(H,Z+1/2),τ(a1(−n1)a2(−n2)···a m(−n m))=(τa1)(−n1)(τa2)(−n2)···(τa m)(−n m).Let o(τσ)=N.We decompose H into eigenspaces with respect to theτσandτas follows:H=⊕r∈Z/N Z H r∗(7.4)H=⊕r∈Z/N0ZH r(7.5) where H r∗={v∈H|τσv=e2πir/N v},and H r={v∈H|τv=e2πir/N0v}.Let l0=dim H0∗.As before we need to consider two separate cases:l0is even or odd. If l0=2k0for some nonnegative integer k0,we haveH0∗=k0i=1C h i+k0 i=1C h∗iwith(h i,h j)=(h∗i,h∗j)=0,(h i,h∗j)=δi,j.Let l r=dim H r∗with r=0.If r=N−r,weﬁx bases b r,1,b r,2,···b r,lr∈H r∗and b∗r,1,b∗r,2,···b∗r,lr∈H(N−r)∗such that(b r,i,b∗r,j)=(b∗r,j,b r,i)=δi,j.If r=N−r,let{c1,c2,···c lN 2∗.Then M= N−1r=1Λ[b(−n)|n∈r2)-module so that for u∈H r∗, Y M(u(−1N +Zu(n)z−n−1/2(see[Li2]).Note that b r,i(n)acts as∂∂c i(−n)if n is positive andacts as multiplication by c i(n)if n is negative.Also,h i(n)acts as∂∂h i(−n)if n is positive,and acts as multiplication by h∗i(n)if nis nonnegative.One can easily calculate thatΩ(M)=Λ[h∗i(0)|h∗i∈H0∗,i=1,2,···k0]. So dimΩ(M)=2k0.Proposition7.4.If dim H0∗=l0=2k0then M= N−1r=1Λ[b(−n)|n∈r2)-module.Proof:As in the proof of Proposition7.2,it is suﬃcient to show that dim Aτ(V(H,Z+ 1N−1 z1+m a(z)b=∞l=0 r2l a(−m−12)).So using the same calculation done in Proposition7.2,we conclude that Aτ(V)is spanned byh1(−1/2)s1···h k0(−1/2)s k0h∗1(−1/2)t1···h∗k(−1/2)t k0with s i,t i=0,1.Hence dim Aτ(V(H,Z+1N+Z,1≤r≤N−1,n>0] are irreducibleτ-twisted V(H,Z+1∂e(−n)if n>0andas multiplication by e(n)if n≤0.The proof of Proposition7.4gives Proposition7.5.If dim H0∗=2k0+1is odd,V(H,Z+1[DVVV]R.Dijkgraaf,C.Vafa,E.Verlinde and H.Verlinde,The operator algebra of orbifold models,Comm.Math.Phys.123(1989),485-526.[DHVW]L.Dixon,J.Harvey,C.Vafa and E.Witten,Strings on orbifolds,Nucl.Phys.B261(1985),651;II,Nucl.Phys.B274(1986),285.[D] C.Dong,Twisted modules for vertex algebras associated with even lattice,J.of Algebra165(1994),91-112.[DL1] C.Dong and J.Lepowsky,Generalized Vertex Algebras and Relative Vertex Operators,Progress in Math.,Vol.112,Birkh¨a user Boston,1993.[DL2] C.Dong and J.Lepowsky,The algebraic structure of relative twisted vertex operators,J.Pure and Applied Algebra110(1996),259-295.[DLM1] C.Dong,H.Li and G.Mason,Regularity of rational vertex operator algebras, Adv.Math.132(1997),148–166.[DLM2] C.Dong,H.Li and G.Mason,Twisted representations of vertex operator alge-bras,Math.Ann.310(1998),571–600.[FFR]Alex J.Feingold,Igor B.Frenkel and John F.X.Ries,Spinor Construction of Vertex Operator Algebras,Triality,and E(1)8,Contemporary Math.121,1991.[FHL]I.Frenkel,Y.Huang and J.Lepowsky,On axiomatic approaches to vertex oper-ator algebras and modules,Mem.Amer.Math.Soc.1041993.[FLM1]I.B.Frenkel,J.Lepowsky and A.Meurman,A natural representation of the Fischer-Griess Monster with the modular function J as character,Proc.Natl.A81(1984),3256-3260.[FLM2]I.B.Frenkel,J.Lepowsky and A.Meurman,Vertex operator calculus,in:Math-ematical Aspects of String Theory,Proc.1986Conference,San Diego.ed.byS.-T.Yau,World Scientiﬁc,Singapore,1987,150-188.[FLM3]I.B.Frenkel,J.Lepowsky and A.Meurman,Vertex Operator Algebras and the Monster,Pure and Applied Math.,Vol.134,Academic Press,1988.[FZ]I.Frenkel and Y.Zhu,Vertex operator algebras associated to representations of aﬃne and Virasoro algebras,Duke Math.J.66(1992),123-168.[KW]V.Kac and W.Wang,Vertex operator superalgebras and representations,Con-tem.Math.,AMS Vol.175(1994),161-191.[Le1]J.Lepowsky,Calculus of twisted vertex operators,Proc.Natl.Acad A 82(1985),8295-8299.。

Interactions of chiral two-forms

a rXiv:h ep-th/991119v221D ec1999ULB-TH-99-28/hep-th/9911109November 1999Interactions of chiral two-forms 1XAVIER BEKAERT Abstract Two issues regarding the interactions of the chiral two-forms are reviewed.First,the problem of constructing Lorentz-invariant self-couplings of a single chiral two-form is investigated in the light of the Dirac-Schwinger condition on the energy-momentum tensor commutation relations.We show how the Perry-Schwarz condition follows from the Dirac-Schwinger criterion and point out that consistency of the gravitational coupling is automatic.Secondly,we study the possible local deformations of chiral two-forms.This problem reduces to the study of the local BRST cohomological group at ghost number zero.We proof that the only consistent deformations of a system of free chiral two-forms are (up to redeﬁnitions)deformations that do not modify the abelian gauge symmetries of the free theory.The consequence of this result for a system consisting of a number of parallel M5-branes is explained.Mathematics Subject Classiﬁcation:81T30,83C99,81T70Keywords:Chiral p-forms,Poincar´e invariance,BRST Cohomology,M5-brane 1Introduction:the chiral two-form and the M5-brane In the limit where bulk gravity decouples,the M5-brane is described by a six-dimensional ﬁeld theory.Its bosonic sector includes,besides the ﬁve scalar ﬁelds which describe the position of the brane in transverse space,a chiral two-form (i.e.a two-form the ﬁeld strength of which is self dual).A single M5-brane with strong classical ﬁelds is well understood;its Lagrangian is described in [1].For a long time,the obstacle to get such an action was due to the presence of the chiral two-form.When one tries to tries to incorporate the self-duality condition into an action a problem arises with preserving Lorentz invariance.There is a simple non-manifestly Lorentz-invariant free Lagrangian which has been given in [2],and which generalizes the Lagrangian of [3]for chiral bosons (the couplings to gravitation are included).As a non linear generalization,a Born-Infeld-like action for for a self-interacting chiral two-form was proposed in [4].The couplings to gravitation were added in a second paper [5].A manifestly covariant formulation of chiral p -forms has been developed in an interesting series of papers [6].This formulation is characterized by the presence of an extra ﬁeld and an extra gauge invariance.This extraﬁeld occurs non-polynomially in the action,even for the free chiral p -forms.It has been shown to be equivalent to the non-manifestly covariant treatment of [2]in Minkowski space [6].The question of Lorentz-invariant self-couplings (as well as consistent self-couplings in an external gravitational background)for a single chiral 2-form is the subject of the section 2.We show that this question can be handled by means of the Dirac-Schwinger condition on the commutation relations of the components of the energy-momentum tensor [7].This condition leads directly to the diﬀerential equation obtained in [4]and implies automatically consistency of the gravitational coupling [8].When several,say n ,M5-branes coincide,little is known pactifying one dimension of M-theory on a circle yields type IIA string theory.If the branes are longitudinal to the compact direction,the M5-branes appear as a set of coinciding D4-branes which are quite well understood.Their dynamics is governed by a ﬁve-dimensional U (n )Born-Infeld theory which,ignoring higher derivative terms,isanordinary U(n)non-abelian gauge theory.Turning back to the eleven dimensional picture,this suggests that a non-abelian extension of the chiral two-form should exist.Deformations of a system of N free chiral two-forms in six dimensions are only considered(N is expected to grow like n3).We ignore the fermions and the scalarﬁelds as we believe that they will not modify our conclusions3.The analysis,outlined in the section3,shows that all continuous and local deformations of the free chiral2-forms does not modify the gauge algebra which remains abelian[11]. As a consequence,no localﬁeld theory continuously connected with the free theory can describe a system of n coinciding M5-branes.This does not exclude non-local deformations of the abelian theory or local lagrangians that cannot be continuously deformed to the free one.2Covariant self interactions of a single chiral two-formThe non manifestly covariant action for a chiral2-form in a gravitational background isS[A ij]= dx0d5xB ij∂0A ij− dx0H(i,j,...=1,...,5)(1)with1B ij=√ǫijmnk B ij B mn.(5)2In order to write the action(1),it is necessary to assume that spacetime has the product form T×Σwhere T is the manifold of the time variable(usually a line).Of course,a spatial coordinate could equivalently play the rˆo le of the time variable,as in[4].Since B ij is gauge-invariant and identically transverse(∂i B ij=0),the action(1)is manifestly invariant under the usual gauge transformationsδΛA ij=∂iΛj−∂jΛi.Inﬂat space,it is also invariant under Lorentz transformations,but these do not take the usual form[2].We deﬁne the exterior form B to be the(time-dependent)spatial2-form with components B ij/√(6)Nand where˜d is the spatial exterior derivative operator.In the case where the second Betti number b2(Σ) of the spatial sections vanishes,this equation implies N(E−B)=˜dm,where m is an arbitrary spatial 1-form.To bring this equation to a more familiar form,one sets m i=A0i.The equations of motion read then F=∗F,where F0ij=˙A ij−∂i A0j+∂j A0i.This is the standard self-duality condition.Alternatively, one may use the gauge freedom to set m=0,which yields the self-duality condition in the temporal gauge(E=B).The brackets of the gauge-invariant magneticﬁelds B ij can be found by follow the Dirac method for constrained systems[12]but one may shortcut the whole procedure and directly read the brackets from the kinetic term of the action(1)[B ij(x),B mn(x′)]=12g B ij B ij,y2=1g,f1=∂1f,f2=∂2f.(10)Then,the condition(8)yields tof21+y1f1f2+(13!ǫijklm F Aklm,F A ijk=∂i A A jk+∂j A A ki+∂k A A ij.(13)The action S0is invariant under the following abelian gauge transformationsδΛA A ij=∂iΛA j−∂jΛA i.(14) This set of gauge transformations is reducible because ifΛA i=∂iǫA,the variation of A A ij is0.Our strategy for studying the possible local deformations of the action(12)is based on the observation that these are in bijective correpondence with the local BRST cohomological group H0,6(s|d)[15],where s is the BRST diﬀerential acting on theﬁelds,the ghosts,and their conjugate antiﬁelds,d is the ordinary space-time exterior derivative and the upper indices refer to ghost number and form degree resp.By applying the standard method of the antiﬁeld formalism[16],one can introduce the ghosts C A i,the ghosts of ghostsηA,and the antiﬁelds A∗Aij,C∗Ai andη∗A.Their respective parity,ghost number, antighost number are listed in following tableantigh gh00C A i0102A∗Aij1-100η∗A3-31-δ-1111antiﬁelds,a=a0.These deformations contain only strictly gauge-invariant terms,i.e.,polynomials in the abelian curvatures and their derivatives(Born-Infeld terms are in this category)as well as Chern-Simons terms,which are(oﬀ-shell)gauge-invariant under the abelian gauge symmetry up to a total derivative. An example of a Chern-Simons term is given by the kinetic term of(12),which can be rewritten as F∧∂0A.5Second,one may deform the action and the gauge transformations while keeping their algebra invariant.In BRST terms,the corresponding cocycles involve(non trivially)the antiﬁelds A∗Aij but not C∗Ai orη∗A.Finally,one may deform everything,including the gauge algebra;the corresponding cocycles involve all the antiﬁelds.Reformulating the problem of deforming the free action(12)in terms of BRST cohomology enables one to use the powerful tools of homological algebra.Following the approach of[17],we have completely worked out the BRST cohomogical classes at ghost number zero.In particular,we have established that one can always get rid of the antiﬁelds by adding trivial solutions[11](The complete proof will be given in[18]).This result can be straightforwardly generalized for a system of chiral p-forms in 2p+2dimensions6.In other words,the only consistent local deformations of a system of free chiral p-forms are(up to redeﬁnitions)deformations that do not modify the abelian gauge symmetries of the free theory.These involve the abelian curvatures or Chern-Simons terms.There are no other consistent, local,deformations(This completely justify the assumptions made for the dependence of H on the A ij through the B ij in the section2).4AcknowledgmentsI am grateful to the respective organizers of the meeting“Quantum aspects of gauge theories,supersym-metry and uniﬁcation”and the“9th Midwest Geometry Conference”for their very enjoyable conferences.I would like to thank my collaborators on the works which were the sources of this talk,M.Henneaux and A.Sevrin.This work has been supported in part by the“Actions de Recherche Concert´e es”of the “Direction de la Recherche Scientiﬁque-Communaut´e Fran¸c aise de Belgique”and by IISN-Belgium. The author is“Chercheur I.I.S.N.”(Belgium).References[1]P.Pasti,D.Sorokin and M.Tonin,Phys.Lett.B398(1997)41,hep-th/9701037;M.Aganagic,J.Park,C.Popescu,J.H.Schwarz,Nucl.Phys.B496(1997)191,hep-th/9701166;I.Bandos,K.Lechner,A.Nurmagambetov,P.Pasti,D.Sorokin and M.Tonin,Phys.Rev.Lett.78(1997)4332, hep-th/9701149[2]M.Henneaux and C.Teitelboim,Phys.Lett.B206(1988)650;M.Henneaux and C.Teitelboim,Consistent Quantum Mechanics of Chiral p-Forms,in Proc.Quantum Mechanics of Fundamental Systems2,Santiago1987,Plenum Press(New York1989).[3]R.Floreanini and R.Jackiw,Phys.Rev.Lett.59(1987)1873.[4]M.Perry and J.H.Schwarz,Nucl.Phys.B489(1997)47-64,hep-th/9611065[5]J.H.Schwarz,Phys.Lett.B395(1997)191,hep-th/9701008[6]P.Pasti,D.Sorokin and M.Tonin,Phys.Lett.B352(1995)59;Phys.Rev.D52(1995)427.[7]P.A.M.Dirac,Rev.Mod.Phys.34(1962)592;J.Schwinger,Phys.Rev.130(1963)406,800.[8]X.Bekaert and M.Henneaux,Int.J.Th.Phys.38(1999)1161,hep-th/9806062[9]I.R.Klebanov and A.A.Tseytlin,Nucl.Phys.B475(1996)164,hep-th/9604089;S.S.Gubser andI.R.Klebanov,Phys.Lett.B41341,hep-th/9708005[10]M.Henningson and K.Skenderis,JHEP9807(1998)023,hep-th/9806087[11]X.Bekaert,M.Henneaux and A.Sevrin,Phys.Lett.B468(1999)228,hep-th/9909094[12]P.A.M.Dirac,Lectures on Quantum Mechanics,Academic Press(New York:1964);M.Henneauxand C.Teitelboim,Quantization of Gauge Systems,Princeton University Press(Princeton:1992).[13]C.Teitelboim,Ann.Phys.(N.Y.)79(1973)542;The Hamiltonian Structure of Spacetime,in GeneralRelativity and Gravitation:One Hundred Years After the Birth of Albert Einstein(A.Held ed.), Plenum Press(New York:1980);C.Teitelboim,Ph.D Thesis(Princeton:1973).[14]M.Hatsuda,K.Kamimura and S.Sekiya,hep-th/9906103.[15]G.Barnich and M.Henneaux,Phys.Lett.B311(1993)123,hep-th/9304057[16]I.A.Batalin and G.A.Vilkovisky,Phys.Lett.B102(1981)27;Phys.Rev.D28(1983)2567;Phys.Rev.D30(1984)508[17]G.Barnich,F.Brandt and M.Henneaux,Comm.Math.Phys174(1995)93,hep-th/9405194;M.Henneaux,Phys.Lett.B368(1996)83,hep-th/9511145;M.Henneaux,B.Knaepen and C.Schomblond,Comm.Math.Phys.186(1997)137,hep-th/9606181;M.Henneaux and B.Knaepen, Phys.Rev.D56(1997)6076,hep-th/9706119;Nucl.Phys.B548(1999)491,hep-th/9812140;hep-th/9912052[18]X.Bekaert,M.Henneaux and A.Sevrin,in preparation[19]X.Bekaert,M.Henneaux and A.Sevrin,hep-th/9912077Adress:Physique Th´e orique et Math´e matique,Universit´e Libre de Bruxelles,Campus Plaine C.P.231B-1050Bruxelles,Belgiumxbekaert@ulb.ac.be6。

基于“OBE”理念的团队科研驱动式教学模式在研究生课程教学中的创新与实践

基于“OBE”理念的团队科研驱动式教学模式在研究生课程教学中的创新与实践王顺宏，雷刚，李杰，潘乐飞，赵久奋（火箭军工程大学，陕西西安710025）一、绪论钱伟长院士曾经说过：“大学必须拆除教学与科研之间的高墙，教学没有科研做底蕴，就是一种没有观点的教育、没有灵魂的教育。

”因此必须将教学与学生科研能力的培养结合起来，这一点对于研究生教学尤其重要。

我校“飞行动力学与制导”课程是大学在新一轮研究生学科专业调整后航空宇航科学与技术学科新设立的一门专业基础课程，其主要课程内容整合原先“飞行器再入动力学与制导”“导引技术”“高超声速飞行器动力学与控制”等课程的部分内容，内容体系涵盖飞行动力学与制导方向主要专业课内容，因此提高该课程的教学效果对本学科方向研究生培养质量有重要作用。

该课程集中运用了飞行器原理、空气动力学、制导与控制原理等航空航天类课程的基础理论和其他先修课程，信息量大，涉及面广，公式多而复杂。

目前该类课程的授课大多秉承传统授课方式，使用固定教材，采用讲授为主，结合少量实验和习题，对学习效果的考核评价亦多采用“平时成绩”+“期终考试”的模式，用这种考核评价模式对教学目标是否达到进行检验，也仅停留在表面，即只能考查对基本原理、概念和方法的掌握，而不能对学生通过该课程的学习，学术科研能力提高进行考核，多年的教学经验已经表明，学生虽然在考试中得到及格甚至良好的成绩，但由于缺乏实践操作和动手能力的培养，那些突击学到的知识在短时间内就很快被遗忘。

OBE（Outcome Based Education）是世界一流大学普遍采用的教学模式，它在提高教学质量和课程评价方面的有效性已经过长期实践和证明，OBE教学模式的理念在于“教”与“学”双方在开始就必须非常清楚地明确该课程的ILOS（Intended Learning Outcomes，即预期学习成果），要根据学生的需求和ILOS的标准来设计所有的教学过程。

考虑到研究生基础理论相对扎实、课程安排相对灵活，方便开展小班教学和一对一指导，且背负科研学术论文发表压力，对科研学术能力提高期望较高，学习主动性较高等现实状况，因此教学组在研究生课程“飞行动力学与制导”课程教学中采用了基于“OBE”理念的团队科研驱动式教学模式，致力将教学过程由传统的理论摘要：文章针对研究生“飞行动力学与制导”课程特点，着眼提高自学能力和科研创新能力，引入基于“OBE”理念的团队科研驱动式教学模式改革。

《初中趣味英语》课件

Grammar in context
Unit 3: Exploring the World of Sports
Sports related vocabulary and expressions
Listening comprehension activities
Grammar review and practice
Sports related vocabulary and expressions
Listening comprehension activities
Grammar review and practice
Utilizing PowerPoint presentations to corporate visuals, audio, and interactive elements for a more engaging learning experience
Multimedia Approach
Implementing practical and meaningful tasks that require students to actively participate and apply their language skills
Task Based Learning
Encouraging students to work together in groups to complete tasks, share ideas, and learn from each other
Collaborative Learning
Introducing innovative and engaging activities such as role plays, games, and simulations to make learning more enjoyable and effective

浙江省网络课程平台-跨文化交际题库(附答案)

Course11.Intercultural competence is the ability to communicate effectively and appropriately in intercultural situations based on one’s intercultural knowledge, _______ and attitudes.【skills】2.Intercultural competence is the ability to communicate effectively and appropriately in intercultural situations based on one’s intercultural knowledge, skills and__________.【attitudes】3.Intercultural competence is the ability to communicate effectively and _____in intercultural situations based on one’s int ercultural knowledge, skills and attitudes.【appropriately】4.Some scholars classify four kinds of intercultural communication. They are ______, interethnic communication, interregional communication and interracial communication.【Intelnational communication】5.Intercultural communication involves interaction between people whose cultural and _________symbol systems are distinct enough to alter the communication event.【perception】1.Taking pictures without asking is considered as violation of privacy in a certain cultural groups.【√】2.Communication barriers are caused by the same communication having different meaning in different cultures.【√】3.International communication takes place between such groups as African Americans and Latin Americans.【×】4.Different people may have different interpretations about what we say and behave because we have different values.【√】5.Intercultural communication involves interaction between people whose cultural perceptions and symbol systems are similar to alter the communication event.【×】Course21.A person’s culture is ___________.【learned】2.Which of the descriptions below is NOT an accurate representation of the relationship between communication and culture?【. Culture and communication are not really related to each other】3.People learn the lessons of culture through __________.【.proverbs myths mass medla all the above】4.According to Edward Tylor, which one of the following is not included by the complex term of culture?【institution】1.The SPEAKING model is a model to analyze spoken and written texts.【×】2.Culture is transmitted from generation to generation to ensure that its crucial messages and elements are passed on.【√】3.Communication can simply refer to the act and process of sending messages among people to connect and interact with other human beings.【×】4.Failure in interaction arose because people in different cultures may choose similar rules and genres to conduct the dialogue.【×】5.Culture is dynamic, which is ongoing and subject to change.【√】6.Feedback is essential to good communication.【√】Course31.In China, as in other Asian societies that have accepted the teaching of Confucius, people are believed to be basically good, yet the ________has a lot of impact on them.【environment】2.Periodic increases in yang are accompanied by corresponding decreases in yin─this followed by an __________ cycle in which yin increases while yang decreases.【opposite】3.People who have the Taoist world view hold that the universe is best seen as an infinite system of elements and_______ in balanced dynamic interaction. Further, two of the forces present in this universe are good and evil.【forces】4.In the Bible story of Adam and Eve, God threw them out of the Garden of Eden because they ate the fruit from ___________.【the tree of knowledge】5.The traditional Western belief about human nature is that human beings are basically ______________.【evil】6.According to Christian teaching, all humans have been born with ___________.【original sin】1.Westerners don’t think we can perfect our nature even if we keep on doing good things.【×】2.Much of Europe also have a good/evil approach to human nature. They believe that while we might be born evil, through learning and education we can become good.【√】3.The traditional Western belief about human nature is that human beings are basically evil.【√】Intercultural conflicts are often caused by differences in value orientations.【√】5.People who have the Taoist world view hold that the universe is best seen as an infinite system of elements and forces in balanced dynamic interaction.【√】6.British anthropologist, Florence Kluckhohn, identified five orientations, five categories of beliefs and behaviors that are universal.【×】7.Buddhism maintains that people are born evil.【×】8.As a result of the rise of humanism in the West, the basic belief has changed to one of seeing humans as a mixture of good and evil. Everybody has both good side and bad side.【×】9.Yang and yin are cyclic; they are regularly repeated; they go through natural periods of balanced increase and decrease.【√】1.Case analysis：Lucia, a Native American college student, when one of her uncles passed away during the first week of school, was expected to participate in family activities. She traveled out of state with her family to his home, helped cook and feed other family members and attended the wake and the funeral. Then her aunt became ill, and she had to care for her. Thus, she missed the first two weeks of school. Some of her professors were sympathetic; others were not. Questions: What was troubling Lucia? Why did she feel troubled?【Lucia feels almost constantly torn between the demands of her collectivistic family and the demands of the individualistic professors and adminstration.】Course41.Being-in-becoming orientation stresses the idea of【development and growth】2.The Arab proverb “The eye cannot rise above the eyebrow” demonstrates ______________attitude.【authoritarian orientation】From former president Lyndon Johnson’s speech“Yesterday is not ours to recover, but tomorrow is ours to win or to lose.”，we can see Americans believe that they can _____________.【control and future】4.Most Latin cultures have the view that the__________ activity is the one that matters the most.【current】5.In present-oriented societies, the past and present often exist side by side. People don’t see any contradiction in that fact. They find it easier to combine __________ and modern ideas and enjoy whatever they have.【tradition】1.In present-oriented societies, the past and present don’t often exist side by side.【错】2.Cultural orientation of controlling nature can also be seen in what is most controversial today, bioengineering and genetic programming.【对】3.The Western approach doesn’t value technology, change and science.【错】4.Many Japanese believe that humans are dominating the nature.【错】5.In Western medicine, the human body is treated as an object that can be studied and then controlled.【对】Course51.C.B. Halverson created a Cultural Context Inventory for us to understand the bigger picture of the context. There are 4 components: Association, Interaction, Temporality, and _____________.【learning】2.America is a _________ culture.【low-context】3.High context communication is communication that places values on implicit information and ___________ cues in order to understand the message. 【nonverbal】4.High context communication is communication that places values on ____________ information and nonverbal cues in order to understand the message.【implicit】5.China is a _________________ culture.【high-context】1.High context communication “Relies heavily on subtle, often non-verbal cues to convey meaning, save face, and maintain social harmony. Communic ators…discover meaning from the context in which a message is delivered.”【对】2.Networking is not important in high-context cultures.【错】3.Examples of high-context cultures include: Asian, African, Arab, Central European, and Latin American countries.【对】4.While high-context communication is a lens in which we can analyze how a message is being interpreted, we can categorize cultures according to generalizations about their communication patterns that influence their communication habits.【错】5.Intercultural communication is a type of communication that shares information across different cultures and societal groups.【对】Course61.In low-context cultures, the communication goal is the exchange of information, ideas, and opinions. When there is a disagreement, it is not _______.【personal】2.Adler & Elmhorts say low context communication “uses ______________ primarily to express thoughts, feelings, and ideas as clearly and logically as possible【language】3.The meaning of a statement in low context culture is in the __________________.【Words spoken】4.Examples of low-context cultures include: American, _______________, Scandinavian, and French.【German】5.Temporality illustrates low-context culture’s approach to time. Events and tasks are _____________and done at specific times【scheduled】6.In low-context cultures, space is compartmentalized and __________ is of the upmost importance.【privacy】7.In low-context cultures, the communication goal is the _______________ of information, ideas, and opinions. 【exchange】8.Low context communication is communicating information ____________ so the message is carried and the information is well determined.【indirectly】1.The low-context association also manifests in a task-oriented mindset where productivity relies on procedures and focusing on the end goal, as well as the identity being found in one’s self and their own accomplishments.【对】2.The meaning of a statement in low context culture is largely in the body language.【错】Course71.Most countries around the world are _______________ .【collectivistic】2.Collectivistic countries typically consist of people that can expect their relatives to look after them in exchange for _____________.【loyalty】3._______________ is the most important in a high power distance society.【respect】The fundamental issue in the Power Distance Dimension is how a society handles _____________ among people.【inequalities】5.Hofstede’s cultural dimensions represent independent preferences for one circumstance or situation over another that distinguishes _____________ from each other.【countries】1.The Power Distance Dimension expresses the degree to which the less powerful members of a society accept and expect that power to be distributed unequally.【对】2.People feel comfortable with inequality and accept the unbalanced distribution of power in a society of high distance power.【对】3.Malaysia is a low distance power culture.【错】4.In societies with lower power distance, people accept a social hierarchy in which everybody has a role to play.【错】5.The first dimension is the Power Distance Dimension.【对】Course81.In the long-term orientation versus short-term orientation dimension, the ________ the score the more emphasis the country shows for long-term orientation.【higher】2.In a masculine society, people regard ______________ as more central to one’s life.【work】3.Countries exhibiting strong Uncertainty Avoidance need ___________ rules.【formal】4.A faminine society at large is more -oriented___________.【concensus】Russia is a typical example of indulgent culture.【错】2.Many countries in Latin America are feminine societies while Spain and Portugal both are categorized as masculine societies.【错】3.Masculinity represents a preference in society for achievement, power, and possessions.【对】4.Planning everything carefully, Germans are not keen on uncertainty.【对】5.Short-term orientation does not emphasize a respect for tradition.【错】6.Indulgence stands for a society that allows relatively free gratification of basic and natural human drives related to enjoying life and having fun.【对】Course91.the four ______ of Asia（亚洲四小龙）【tigers】2.a _________ in the path/way (拦路虎)【lions】3.teach _________ to swim （班门弄斧）【fish】4.Spring up like _________（雨后春笋）【mushrooms】5.One _______ does not make a summer. （独木不成林）【seallow】6.Spend money like _________（挥金如土）【water】7.a _________ movie （黄片）【red】a _________ day （吉日）【white】9.______Fraiday（灾难的一天）【black】10.________ at one’s job （无工作经验）【red】1.Although both dove and 鸽子refer to the same bird, they connotate different meanings.【错】2.The color of red is always associated with something dangerous and threatening in the West.【对】3.猫头鹰in Chinese symbolizes wisdom.【错】4.The number of eight is taken as a lucky number in Chinese as well as English.【错】5.Verbal communication is the communication or exchange of ideas that occurs through written words.【错】6.Like Chinese people, Americans associate the moon with their home.【错】7.A word’s denotation refers to its literal or dictionary meaning.【对】8.A word’s connotation refers to its assciated meaning.【对】9.In verbal communication, we find differences at phonemic, lexical, syntactic, discourse and pragmatic levels between different cultures.【对】10.Words of the same literal meaning from different cultures always have same associated meanings.【错】Course101.The westerners tend to focus on the stages of the discourse as the most crucial __________________.【opening】2._______________ tend to look for the crucial points to occur somewhat later.【the Chinese】3.When we talk to someone who is superior to you, we tend to use______________ , because it’s more indirect and more polite.【logical pattern】4.Discourse patterns greatly depend on ___________________ .【thought pattern】5.Some factors such as ________________, social distance and weight of imposition of a request influence the choice of induction and deduction.【power distance】6.English paragraph development is characterized by _________________, directness, clarity and logic.【linearity】1.We can never take a single word as discourse.【错】2.An English paragraph typically begins with a topic statement.【对】3.Induction is a reasoning process in which particular or minor points move forwards a general or major topic.【对】4.Discourse is naturally spoken or wirtten language in context.【对】Course111.For English native speakers, an apology is necessary whenever inconvenience or offence is made with little consideration of the __________ of the people concerned.【status】2.In English the offer is mostly in the form of a/an【question】3.In Chinese, the proper use of kinship terms to address others, strangers or relatives, can show a person’s _______________, respectfulness and friendliness.【politeness】1.A Chinese name is usually arranged in the order of given name plus surname.【错】2.Some family members might call each other by their first names in English speaking countries.【对】3.The Chinese host makes the offer directly like a command or a statement.【对】1.Li Ming was a Chinese student studying in the US. He was lost on his way to the post office, so when he saw an old lady about the same age as his grandmother, he went to her for help. Li Ming: Excuse me, Granfma, could you tell me how to get to the nearest post office? The lady: Are you talking to me, young man? I’m not your grandma! Question: Was it appropriate for Li Ming to use the kinship term “grandma” as a form of addressing to the lady in this conversation?【It is not acceptable to apply a kinship term to greet a stranger in English. Li Ming made a pragmatic transfer.】Course121.Nonverbal communication is simply known as the “_________ ” language or the “hidden dimension” of communication, however you and I already know all about it!【silent】2.An American poet, Ralph Waldo Emerson, once said “What you do speaks so loudly that I cannot hear what you say”. What we “do” is our ________________.【nonverbal communication】3.Ray Birdwhistell, an American anthropologist, estimates that 65 to 70 percent of the conversation is carried by ________________ communication.【nonverbalcommunication】1.Nonverbal communication can occur without the presence of words.【对】2.Body movements can include posture, gestures, facial expressions, eye contact, and other displays.【对】1.What is kinesics?正确答案:Kinesics, or body language, refers to all nonverbal codes, which are associated with body movements. Body movements can include posture, gestures, facial expressions, eye contact, and other displays.Course131.Time perceptions include: , willingness to wait, _______________ and interactions.【punctuality】2.Hall describes proxemics as having two main types of space: territory and ______________ space.【personal】1.How you are aware of time varies by culture and normative expectations of adherence to time.【对】2.M-Time cultures are typically task-oriented.【对】3.Personal space determines how we interact in the real world. Edward Hall postulates that there are 4 zones: intimate, personal, social, and public.【对】Course151．Racism is associated with oppression and power.【√】2．Most people assume that other people perceive, evaluate, and reason about the world in the same way that they do.【√】3．It is easy to get rid of cultural biases as they are bad.【×】4．People generally perceive their own experiences, which are shaped by their own cultural forces, as natural, human, and universal.【√】5．Racism is the tendency by group in control of _____________ to use it to keep members of groups who do not have access to the same kinds of power at a disadvantage.【power】6．Stereotypes are always negative.【×】7．Ethnocentrism is “the notion that the beliefs, values, norms, and practices of one’s own culture are __________ to those of others.”【正确的，最好的，superior】8．The consequence of stereotyping is that _______________ among the members of any one group may be ignored.【differences】9．Ethnocentrism tends to ____________ cultural differences.【highlight and exaggerate】10．The disadvantage of the human information processing is that it can create some ______________ to intercultural competence.【discrimination and racism obstacles】Course161.At any given moment, one has many ‘ ________ ’ that make up your identity.【facets】2.Mr. Portokalos, the father in the movie My Big Fat Greek Wedding hangs a _______ flag to show the family’s tribute to their heritage.【Greek】3.Cultural differences and distinguishing characteristics can __________________ one culture from another.【differentiate】4.Developing one’s involves learning about __________________ and accepting the traditions, heritage, language, religion, ancestry, aesthetics, thinking patterns, and social structures of a culture.【cultural identity】5.The stage of cultural identity achievement is characterized by a dear, confident acceptance of _____________ and an internalization of one's cultural identity.【oneself】1.During the stage of unexamined cultural identity, one’s cultural characteristics are taken for granted.【√】During the stage of an examiner cultural identity, once cultural characteristics ar e taken for granted.2.When children notice that some of their playmates have different colored skin, they may fear or feel superior to them like adults.【×】even when children may notice that some of their places have different color sche me. as they are not aware of the significance associated with the color scheme, th ey do not fear or feel superior to them.3.In the second stage, for some individuals, there might be a turning point or crucial event leading to this stage.【√】In the second stage,cultural identity research involves a process of exploration and questioning about one's culture and not to learn more about it.And to understand the imp lications of membership in that culture for some individuals,there might be a turning point of crucial event leading to the stage.4.Cultural identity refers to one’s sense of belonging to a particular culture or ethnic group.【√】or i'm American by saying that you have consciously or unconsciously formed your c ultural identity and which re fers to one sense of belonging to a particular culture or et hnic group.5.Cultural identities are static, fixed, and enduring within a changing social context.【×】Cultural identities are dynamic,so your cultural identity exists within a changing soci al context.Course171．Honeymoon phase is followed by adjustment phase when the person feels confused and uncomfortable.【×】2．Culture shock is precipitated by the anxiety that results from losing all our familiar signs and symbols of social intercourse.【√】3．All through life people are confronted with new situations that require them to adjust their thinking and behavior【√】4．Adaptation is a process with ________________________ stages.【identifiable】5．When we are feeling confused, frustrated or tense because of intercultural misunderstandings, please remember that the feelings are _________________ .【only temporary】6．To adapt more successfully, we need to pay attention to differences within thenew culture: avoid making broad ________________ about everybody in the hostculture;【generalize Asians】7．To adapt more successfully, please be alert to signs of adaptation stress: health problems; loss of self-confidence; loneliness; sense of loss; severe homesickness; withdrawal from social contacts; negative feelings; behaving more_________________ than usual.【aggressively】8．The third stage of culture shock is characterized by gaining some _________________ of the new culture.【understanding】9．To adapt more successfully, please seek _____________ experiences within the new culture: if you like to watch football, watch football with people from the new culture; if you enjoy music, enjoy it with people form the new culture【positive】10．To adapt more successfully, please use your cross-cultural experience to increase your skills: Notice and ______________ the communication styles of people from the new culture【imitate】Course181．Sometimes reentry shock is even more difficult than culture shock.【√】2．When people return home to their original cultural contexts, the same processof adaptation occurs and may again involve culture shock.【√】3．Reentry shock can be depicted by the U-curve model.【×】4．The W-curve pattern suggests that when we return home, we must proceedthrough the four stages of the U-curve pattern once again.【√】5．The person who return home is the same person who left home.【×】6．There are two fundamental differences between the first and second U curves, related to issues of personal change and expectations. In the reentry phase, the sojourner has _______________ through the adaptation process.【has changed through】7．International students who return home also talk about how their friends and families expect them to be a little different (more educated) but basically the ___________________as before they went off to school.【same】8．In a new environment, one will lose all the familiar ____________ from his home culture.【similar cues from】9．Experiencing culture shock has a strong ___________________ to make people be multicultural or bicultural.【potential】10．There are two fundamental differences between the first and second U curves, related to issues of personal change and expectations. In the initial curve or phase, the sojourner is fundamentally _________________ and is experiencing new culturalcontexts.【unchanged】11．Experiencing culture shock gives the sojourners a chance to learn moreabout ___________________ and at the same time, other cultures.【themselves】12．The new environment makes demands for which one has no _________________responses.【ready-made】。

2022年考研考博-考博英语-湖南师范大学考试全真模拟易错、难点剖析AB卷(带答案)试题号：78

2022年考研考博-考博英语-湖南师范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A magician’s talk creates a（）of attention so that people do not see how he does his tricks. 问题1选项A.vacuumB.concentrationC.divisionD.diversion【答案】D【解析】vacuum真空；concentration集中；division区分；diversion转移，分散注意力。

句意：魔术师的谈话转移了人们的注意力，使人们看不见他是如何表演魔术的。

选项D符合句意。

2.单选题Originally designed as work clothes for miners, jeans are now worm by all segments of society, their appeal ______ by their comfort and affordability.问题1选项A.overwhelmedB.corrodedplicatedD.broadened 【答案】D【解析】考查动词辨析。

A选项overwhelmed“（感情或感觉）充溢，难以禁受；击败，征服”；B选项corroded“腐蚀，侵蚀”；C选项complicated“使复杂”；D选项broadened“扩宽，扩大”。

句意：牛仔裤最初是为矿工设计的工作服，现在受到社会各阶层的追捧，它的吸引力因为舒适和实惠的特点而______。

这里要填的是牛仔裤的吸引力因为舒适和实惠的特点而怎么样，D选项broadened“扩宽，扩大”符合题意，也对应前面牛仔裤受欢迎的特点。

因此D选项正确。

3.单选题The elegant decorations （）the gym into a starlit ballroom.问题1选项A.transplantedB.transferredC.transcendedD.transformed【答案】D【解析】transplant移植, 迁移；transfer转让, 移交；transcend胜过, 超越；transform改变, 转换。

Local systems of vertex operators, vertex superalgebras and modules

−1 δ = Resz1 z0
(1.2)
z − z1 z1 − z −1 a(z1 )b(z ) − (−1)|a(z )||b(z )| z0 δ b(z )a(z1 ) . (1.3) z0 −z0
This is our desired “product” formula. Noticing that the expression of the right hand side of (1.3) is not well deﬁned for arbitrary a(z ), b(z ) ∈ (EndM )[[z, z −1 ]], we restrict our attention to the subspace F (M ) consisting of those formal series a(z ) satisfying the truncation condition, i.e., a(z )u ∈ M ((z )) for any u ∈ M ; note that we must have A ⊆ F (M ). Then F (M ) is a subspace on which the right-hand side of (1.3) is well d acts. Let I (z ) = idM be the identity operator of M . Then deﬁned and on which D = dz we obtain a quadruple (F (M ), I (z ), D, Y ). But (F (M ), D, I (z ), Y ) is not a vertex algebra. To a certain extent, vertex algebras look like commutative associative algebras with identity because of the “commutativity” and the “associativity.” Just as there is no canonical largest commutative associative algebra associated to a vector space, there is no (canonical) universal or largest vertex algebra associated to M , but there are maximal vertex algebras inside the space F (M ), what we call “local systems of vertex operators.” 3

蓝莓花色苷聚电解质复合物制备及降脂活性比较

蓝莓花色苷聚电解质复合物制备及降脂活性比较尹朝春1，李环通2，许泽文3，陈丹妮1，王赛男1，肖苏尧1*（1.华南农业大学食品学院，广东省功能食品活性物重点实验室，广东广州 510642）（2.广东茂名农林科技职业学院食品工程系，广东茂名 525024）（3.华润怡宝饮料（中国）有限公司，广东深圳 518055）摘要：为提高蓝莓花色苷的稳定性和活性，该研究利用大豆分离蛋白和阿拉伯胶构建大豆分离蛋白/阿拉伯胶聚电解质负载蓝莓花色苷体系并进行表征，并对复合物的细胞降脂活性进行了较研究。

在超声功率180 W、壁材质量比为10:4、壁芯质量比为10:1、包埋时间为1.0 h的条件下，得到包埋率为66.05%，ζ-电位为1比9.00 mV，平均粒径为2.53 μm的不规则球状物，此条件下制备的复合物，在模拟胃液和肠液中2 h的释放率为71.05%、61.04%，比游离花色苷降低了19.16%、30.50%。

通过HepG2细胞模型，检测TG、TC、SOD、MDA含量，结果发现，复合物的降脂活性优于游离花色苷，当花色苷质量浓度在50 μg/mL时，复合物组的TC、TG、MDA含量为0.19 mmol/g prot、0.21 mmol/g prot、10.58 nmol/mg prot，比游离花色苷组下降9.70%、14.21%、17.12%；SOD酶活力为25.25 U/mg prot，比游离花色苷组提高9.54%。

表明大豆分离蛋白和阿拉伯胶可高效结合花色苷，形成稳定的复合物，并具有良好的体外缓释效果，且复合物降脂活性高于游离花色苷。

该研究可为后续开发降脂产品提供理论依据。

关键词：蓝莓花色苷；聚电解质；大豆分离蛋白；阿拉伯胶文章编号：1673-9078(2024)04-35-45 DOI: 10.13982/j.mfst.1673-9078.2024.4.0321Preparation and Comparative Analysis of Lipid-lowering Activity ofBlueberry Anthocyanin Polyelectrolyte ComplexesYIN Zhaochun1, LI Huantong2, XU Zewen3, CHEN Danni1, W ANG Sainan1, XIAO Suyao1*(1. College of Food Sciences, Guangdong Provincial Key Laboratory of Nutraceuticals and Functional Foods, SouthChina Agricultural University, Guangzhou 510642, China)(2. Guangdong Maoming Agriculture & Forestry Technical College, Department of Food Engineering, Maoming525024, China)(3. China Resources C’estbon Beverage (China) Co. Ltd., Shenzhen 518055, China)Abstract: To improve the stability and activity of blueberry anthocyanins, soy protein isolate, and acacia gum were used to construct and characterize soy protein isolate/acacia polyelectrolyte loaded with blueberry anthocyanins. The lipid-lowering activities of the complexes were compared. Under the conditions of ultrasonic power of 180 W, a wall material quality ratio of 10:4, wall core quality ratio of 10:1, and embedding time of 1.0 h, irregular spherical particles with an embedding rate of 66.05%, a ζ-potential of 1 to 9.00 mV, and an average particle size of 2.53 μm were obtained. The 2 h-release 引文格式：尹朝春,李环通,许泽文,等.蓝莓花色苷聚电解质复合物制备及降脂活性比较[J].现代食品科技,2024,40(4):35-45.YIN Zhaochun, LI Huantong, XU Zewen, et al. Preparation and comparative analysis of lipid-lowering activity of blueberry anthocyanin polyelectrolyte complexes [J]. Modern Food Science and Technology, 2024, 40(4): 35-45.收稿日期：2023-03-16基金项目：广东省基础与应用基础研究基金面上项目(2022A1515010907)作者简介：尹朝春（1998-），女，在读硕士研究生，研究方向：天然产物活性功能，E-mail：通讯作者：肖苏尧（1974-），女，博士，副教授，研究方向：天然产物活性功能，E-mail：35蓝莓花色苷（Blueberry Anthocyanin，BBA）具有包括抗氧化、降脂、降血糖、抗炎、促进益生菌增殖、抗肿瘤、保护视网膜等多种生物活性，在保健食品开发、化妆品行业具有广阔的应用前景[1] 。

vertex attributes

vertex attributesVertex Attributes: Enhancing the Power of GraphsIntroductionGraph theory is a powerful mathematical tool that has found applications in various fields such as computer science, social networks, genetics, and transportation systems. Graphs consist of vertices (nodes) and edges (connections) that represent relationships between entities. While edges define the connections between vertices, vertex attributes provide additional information about individual vertices, making graphs more expressive and meaningful. This article explores the concept of vertex attributes, their importance, and their applications in different domains.Understanding Vertex AttributesVertex attributes refer to the characteristics or properties associated with each vertex in a graph. These attributes can be numerical or categorical, representing different types of information about the vertices. For example, in a social network graph, vertex attributes can include attributes such as age, gender, location, and occupation. In a transportation network, vertex attributes might include attributes such as the average daily traffic volume or the type of road.Importance of Vertex AttributesVertex attributes significantly enhance the power and flexibility of graph analysis. Here are some key reasons why vertex attributes are important:1. Enriched Information: Vertex attributes provide additional information about vertices, enabling a more comprehensive understanding of the graph's structure and dynamics. By incorporating attributes, graphs become more than just a collection of connections, but also a representation of the attributes associated with the entities being modeled.2. Contextual Analysis: Vertex attributes allow for context-aware analysis. By considering the attributes associated with vertices, researchers can gain insights into the specific characteristics of different entities within a graph. For example, analyzing attributes in a social network graph can revealpatterns of behavior among different age groups or genders.3. Enhanced Visualization: Vertex attributes can be leveraged to create visually appealing and informative graph visualizations. By mapping attributes to different visual properties such as color, size, or shape, it becomes easier to identify specific attributes within a large or complex graph. This improves interpretability and aids in effective data communication.4. Feature Extraction: Vertex attributes can be used as input features in machine learning and data mining algorithms. By leveraging attribute information, it becomes possible to develop predictive models or classification techniques that utilize the attributes associated with vertices to make accurate predictions or identify patterns.Applications of Vertex AttributesThe usage of vertex attributes is extensive across various domains. Here area few examples that highlight the practical applications:1. Social Networks: Vertex attributes are fundamental in social network analysis. They help identify influential individuals, detect communities, and predict user behavior based on attributes such as interests, location, and profession. This information is invaluable for targeted marketing, recommendation systems, and understanding social dynamics.2. Biological Networks: In genetics, vertex attributes can represent gene expression levels, DNA sequence properties, or protein interactions. Analyzing these attributes can aid in understanding biological processes, identifying disease markers, and predicting protein functions.3. Transportation Networks: In transportation systems, vertex attributes can include traffic flow, road accessibility, or pavement conditions. By incorporating these attributes, it becomes possible to optimize routes, identify congestion hotspots, and plan infrastructure improvements.4. Financial Networks: In finance, vertex attributes can represent characteristics such as credit rating, investment portfolio, or transaction history. Analyzing these attributes can facilitate risk assessment, fraud detection, and portfolio optimization.Challenges and Future DirectionsAlthough vertex attributes offer various advantages, there are challenges associated with their usage:1. Data Quality: Ensuring the accuracy, consistency, and completeness ofvertex attribute data can be challenging. In real-world scenarios, data may be missing or contain errors, hampering accurate analysis and interpretation. 2. Scalability: As graph sizes increase, managing and processing extensive vertex attribute data becomes computationally expensive. Developing efficient algorithms capable of handling large-scale attribute-rich graphs is necessary for scalable analysis.Looking ahead, here are some potential future directions for the study and utilization of vertex attributes:1. Attribute Inference: Developing algorithms or techniques to infer missing attributes from available data can help mitigate the data quality challenge and enhance the usability of graphs with incomplete information.2. Dynamic Attribute Analysis: Extending vertex attribute analysis to incorporate temporal or evolving attributes will enable the study of changing network behavior over time. This can be particularly beneficial in domains such as social networks or transportation systems where attributes can change dynamically.ConclusionVertex attributes play a vital role in enhancing the power of graph analysis. They provide additional information about individual vertices, enabling a deeper understanding of graph structures, context-aware analysis, and data-driven decision-making. The practical applications of vertex attributes span across various domains, facilitating the analysis of social, biological, transportation, and financial networks. However, the effective usage of vertex attributes requires addressing data quality issues and developing scalable algorithms. The future holds exciting possibilities for leveraging attributes to infer missing information and incorporating temporal dynamics into graph analysis, further elevating the utility of vertex attributes in various fields.。

大学生自我控制量表的修订

大学生自我控制量表的修订谭树华，郭永玉(华中师范大学心理学院，湖北武汉430079)【摘要】目的：修订自我控制量表(SCS)，考察其心理测量学指标。

方法：对799名武汉市大学生进行测查，对量表进行验证性因素分析和信、效度检验。

结果：验证性因素分析的结果显示，SCS的五因素结构拟合较好。

SCS的内部一致性信度为0.862，重测信度为0.850。

以被试的平均学分绩、人际关系满意感、生活满意感、心理健康水平为效标，与SCS的相关分别为0.146；0.280；0.163；0.317。

结论：SCS符合心理测量学的要求；可作为测量我国大学生自我控制能力的工具。

【关键词】自我控制量表；信度；效度中图分类号:R395.1文献标识码:A文章编号:1005-3611(2008)05-0468-03Revision of Self-Control Scale for Chinese College StudentsTAN Shu-hua，GUO Yong-yuSchool of Psychology，Central China Normal University，Wuhan430079，China【Abstract】Objective：To revise the Self-Control Scale(SCS).Methods：Data were collected from a sample of799col-lege students of Wuhan and analyzed by Confirmatory Factor Analysis,reliability test and validity test.Results：The re-sults of Confirmatory Factor Analysis showed that the revised SCS was five-factor construct and had good construct validi-ties.The Cronbach’sαcoefficient of the SCS scale was0.862,and the reliability coefficient of the test-retest stability co-efficient was0.850.The correlation between total score of SCS and that of Grade Point Average(GPA)was0.146;the cor-relation between total score of SCS and that of Interpersonal Satisfaction Scale(ISS)was0.280;the correlation between total score of SCS and that of Life Satisfaction Scale(LSS)was0.163,and the correlation between total score of SCS and that of General Health Questionnaire(CHQ)was0.317.Conclusion：The revised scale of SCS has good psychometric quality and can be used in Chinese college students.【Key words】Self-control Scale；Reliability；Validity人们最健康、最幸福的时候就是自我和环境完全匹配的时候，只是日常生活中个体和环境很难完全匹配，但匹配的程度可以通过调整自我来得到最大化地提升[1]。

1101.4484v1 Some general results on conformal embeddings of affine vertex operator algebras

a r X i v :1101.4484v1[mat h.Q A]24J a n211SOME GENERAL RESULTS ON CONFORMAL EMBEDDINGS OF AFFINE VERTEX OPERATOR ALGEBRAS DRA ˇZEN ADAMOVI ´C AND OZREN PER ˇSE Abstract.We give a general criterion for conformal embeddings of vertex operator algebras associated to aﬃne Lie algebras at arbitrary ing that criterion,we construct new conformal embeddings at admissible rational and negative integer levels.In particular,we con-struct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras.The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.1.Introduction Let U and V be vertex operator algebras of aﬃne type.We say that U is conformally embedded into V if U can be realized as a vertex subalge-bra of V with the same Sugawara Virasoro vector.The most interesting case of conformal embedding is when V is a ﬁnite direct sum of irreducible U –modules.These conformal embeddings are studied in various aspects of conformal ﬁeld theory (cf.[6],[8],[9],[15],[22],[36]),the theory of tensor cat-egories (cf.[21],[28])and in the representation theory of aﬃne Kac-Moody Lie algebras (cf.[12]).The construction and classiﬁcation of conformal embeddings have mostly been studied for simple aﬃne vertex operator alge-bras of positive levels.Some examples of conformal embeddings at rational admissible levels were constructed in [33]and [35].In the present paper we give both necessary and suﬃcient conditions forconformal embeddings at general levels,within the framework of vertex op-erator algebra theory.In this way we will be able to construct new examples of conformal embeddings at rational and negative integer levels.Let us ex-plain our results in more details.Let g be a simple ﬁnite-dimensional Lie algebra and g 0its subalgebra which is a reductive Lie algebra.Let N g (k,0)be the universal aﬃne vertex operator algebra of level k =−h ∨associated to g ,and L g (k,0)its simple quotient.Then g 0generates a subalgebra U g0(resp. U g 0)of N g (k,0)(resp.L g (k,0)).Let ωg 0be the Virasoro vector in U g 0which is a sum of usualVirasoro vectors obtained by using Sugawara construction.Let 2000Mathematics Subject Classiﬁcation.Primary 17B69,Secondary 17B67,81R10.12DRAˇZEN ADAMOVI´C AND OZREN PERˇSEn∈ZL(0)x(−1)1=x(−1)1(x∈W).(1.1)Moreover,the condition(1.1)implies that N g(k,0)contains a singular vector of conformal weight2.If k is a positive integer,then N g(k,0)has a singular vector of conformal weight2if and only if k=1.So we only have conformal embeddings into L g(1,0).But in general we have such singular vectors for other values of k (cf.Remark2.4).Next we apply Theorem1.1in the case when g0is a simple Lie algebra, andg=g0⊕V g0(µ1)⊕···⊕V g(µs),where V g0(µi)is irreducibleﬁnite-dimensional g0-module with highest weightµi,and V g(µi)(i=1,...,s)is orthogonal to g0with respect to the invariant bilinear form on g.Denote by L g0(k′,0)the subalgebra of L g(k,0)generated by g0,where k′=ak and a∈N is the Dynkin index of the embedding g0<g.Note that in the case of general levels,vertex algebra L g0(k′,0)is not necessarily simple.Theorem1.1then implies that the key condition for the conformal em-bedding is that the eigenvalue of the Casimir operator of g0is the same for all V g(µi),i=1,...,s.More precisely,if(µi,µi+2ρ0)0=(µj,µj+2ρ0)0for all i,j=1,...,s, where(·,·)0denotes the suitably normalized invariant bilinear form on g0, then for k,k′such that(µi,µi+2ρ0)03 Our methods enable us to construct all conformal embeddings associated to automorphisms of Dynkin diagrams.Let us illustrate this result by the following table:Table A,g,g0are simple Lie algebras,g0<g.g decomposition of L g(k,0)k′Cℓ−1A2ℓL g0(k′,0)⊕L g(k′,2ω1)2A11DℓL g0(k′,0)⊕L g(k′,ω1)−ℓ+2F4−3D4L g0(k′,0)⊕L g(k′,ω2)⊕−2L(1)g0(k′,ω1)⊕L(2)g(k′,ω1)In order to describe all conformal embeddings for such automorphisms,we include into this table some well-known conformal embeddings at positive integer levels in the cases(A2ℓ,Bℓ),(A2,A1)(cf.[22];see also[4],[37])and the conformal embedding for pair(A2ℓ−1,Cℓ)from[5].But other conformal embeddings at negative integer levels are new.We also apply our methods to aﬃne vertex operator algebras at admissible levels(cf.[1],[3],[26],[27],[32],[33],[34],[37])and get new conformalembedding of L A2(−5/3,0)into L G2(−5/3,0)with decompositionL G2(−5/3,0)=L A2(−5/3,0)⊕L A2(−5/3,ω1)⊕L A2(−5/3,ω2).We also determine the decomposition for the conformal embedding of L Dℓ(−ℓ+3/2,0)into L Bℓ(−ℓ+3/2,0)from[35].We should also mention that our methods give an uniform proof for a family of conformal embeddings which does not require explicit realizations of aﬃne Lie algebras or explicit formulas for singular vectors.In most cases one can check all conditions for conformal embeddings only by using tensor product decompositions ofﬁnite-dimensional modules of simple Lie algebras. The notion of conformal embedding is also closely related to the notion of extension of vertex operator algebra.We note that simple current extensions of aﬃne vertex operator algebras at positive integer levels were studied in [13],[31].We assume that the reader is familiar with the notion of vertex operator algebra(cf.[10],[18],[19]).2.A general criterion for conformal embeddingsIn this section we shall derive a general criterion for the conformal embed-ding of aﬃne vertex operator algebras associated to a pair of Lie algebras (g,g0),where g is a simple Lie algebra and g0its reductive subalgebra.Let g be the simple complexﬁnite-dimensional Lie algebra with a non-degenerate invariant symmetric bilinear form(·,·).We can normalize the form on g such that(θ,θ)=2,whereθis the highest root of g.Letρbe the4DRAˇZEN ADAMOVI´C AND OZREN PERˇSEsum of all fundamental weights for g and let h∨be the dual Coxeter number for g.Assume that g0is a Lie subalgebra of g such that(1)bilinear form(·,·)on g0is non-degenerate and g=g0⊕W is anorthogonal sum of g0and a g0–module W;(2)g0=⊕n i=0g0,i is an orthogonal sum of commutative subalgebra g0,0and simple Lie algebras g0,1,...,g0,n(i.e.,g0is a reductive Lie alge-bra).Let(·,·)g0,i be the non-degenerate bilinear form on g0,i normalized suchthat(θ0,i,θ0,i)g0,i =2,whereθ0,i is the highest root of g0,i.One can showthata i(·,·)g0,i=(·,·)|g0,i×g0,i for certain a i∈N.(For i=0,we can take a0=1).The(n+1)–tuple(a0,a1,···,a n)is called the Dynkin multi-index of embedding g0into g.Let{u i,j}j,{v i,j}j be a pair of dual bases for g0,i such that(u i,j,v i,k)g0,i=δj,k.Let h∨0,i be the dual Coxeter number for g0,i.(For i=0we take h∨0,0=0).Let N g(k,0)be the universal aﬃne vertex operator algebra of level k=−h∨associated to g(cf.[17],[20],[25],[29],[30]).Let N1g(k,0)be the max-imal ideal of N g(k,0),and L g(k,0)=N g(k,0)/N1g(k,0)the corresponding simple vertex operator algebra.Then g0,i generates a vertex subalgebra of N g(k,0)which is isomorphicto N g0,i (k i,0)for k i=a i k.Moreover,g0generates a subalgebra of N g(k,0)which is isomorphic toU g0=⊗n i=0N g0,i(k i,0).Let U g0be the subalgebra of L g(k,0)generated by g0.Then U g0is iso-morphic toU g0=⊗n i=0 L g0,i(k i,0), where L g0,i(k i,0)is a certain quotient of N g0,i(k i,0).Deﬁnition2.1.We say that the vertex operator algebra U g0is conformally embedded into L g(k,0),if U g0is a vertex subalgebra of L g(k,0)with the same Virasoro vector.Letωg be the usual Sugawara Virasoro vector in N g(k,0)and letωg0,i =15 As usual we identify x∈g with x(−1)1∈N g(k,0).Therefore,W can be considered as a subspace of N g(k,0).LetL(n)z−n−2=Y(ωg,z).Theorem2.2.Assume that2(k i+h∨0,i)[[x,u i,j],v i,j](−1)1=k1+h∨0,1+···+k n dim(g0,1)k+h∨.(2.6)So our condition(2.1)from Theorem 2.2implies numerical criterion (2.6).But,in general,(2.6)is only a necessary condition for conformal embeddings.Remark2.4.Assume that the conditions from Theorem2.2are satisﬁed. Then N g(k,0)must have at least one singular vector at conformal weight2. Some explicit formulas for these singular vectors appeared in[1],[2],[32], [33],[34]and recently in[7].We shall now see how the theory presented hereﬁts into some examples.6DRA ˇZEN ADAMOVI ´CAND OZREN PER ˇSE Example 2.5.Let g be a simple complex Lie algebra of type A ,D or E ,h its Cartan subalgebra and ∆its root system.Then g =h ⊕W,W = α∈∆g α.Take now g 0=h in the above construction.ThenL (0)|W ≡Id iﬀk =1or k =−1/2.This leads to the conformal embeddings L g 0(2,0)<L g (1,0), L g 0(−1,0)<L g (−1/2,0).Example 2.7.By using the same method as above we obtain the following conformal embeddings:(1)Let g =sl (ℓ+1,C ),g 0=gl (ℓ,C ).Then we have conformal embeddings:L g 0(1,0)<L g (1,0),L g 0(−ℓ+12,0).(2)Let g =o (2ℓ,C ),g 0=gl (ℓ,C ).Then we have conformal embeddings:L g 0(1,0)<L g (1,0), L g 0(−2,0)<L g (−2,0).Remark 2.8.One can show that in examples presented above the vertex operator algebra L g (k,0)is not a ﬁnite sum of irreducible L g 0(k ′,0)–modules.But it is still possible that there exists certain twisted L g (k,0)–module which is a ﬁnite sum of irreducible twisted L g 0(k ′,0)–modules.In particular,this holds for Z 2–twisted module for L g (1,0)which gives principal realization of level one modules for aﬃne Lie algebras in Example 2.5.It seems that to get conformal embeddings such that L g (k,0)is a ﬁnite sum of irreducible L g 0(k ′,0)–modules,one needs to consider the case when g 0is semisimple.In this article we shall make the ﬁrst step and consider the important case when g 0is a simple Lie algebra.So we shall now assume that g 0is a simple Lie algebra.In the above settings we take n =1,g 0=g 0,1.Set(·,·)0=(·,·)g 0,1,h ∨0=h ∨0,1,a =a 1.Let ρ0be the sum of all dominant integral weights for g 0.7 Assume thatg=g0⊕g1⊕···⊕g s,such that(1)g i=V g0(µi)is irreducibleﬁnite-dimensional highest weight g0-modulewith highest weightµi,(2)g i⊥g0for i=1,...,s(with respect to(·,·)),(3)(µi,µi+2ρ0)0=(µj,µj+2ρ0)0(i,j>0).Then g0generates a subalgebra of N g(k,0)isomorphic to U g0∼=Ng0(k′,0),where k′=ak.Letωg(resp.ωg0)be the Virasoro vector in N g(k,0)(resp.N g(k′,0))obtained by the Sugawara construction.In what follows we choose k such that(µi,µi+2ρ0)0L(0)x(−1)1=x(−1)1(x∈W).As before,let U g0= L g0(k′,0)be the subalgebra of L g(k,0)generated by g0.Now Theorem2.2implies the following result: Corollary2.9.Assume that the above conditions hold.Then L g0(k′,0)is conformally embedded into L g(k,0).The simplicity of the vertex operator algebra L g0(k′,0)will be investigated in the next section.3.On semisimplicity of L g(k,0)as g0–moduleIn this section we give suﬃcient conditions for complete reducibility of L g(k,0)as g0–module.In particular,we obtain the conformal embedding ofsimple vertex operator algebras L g0(k′,0)<L g(k,0).We assume that g0is a simple Lie algebra and that all conditions on embedding g0<g from Section2hold.Furthermore,we assume that g0is theﬁxed point subalgebra of an automorphismσof g of order s+1,and thatV g0(µi)is the eigenspace associated to the eigenvalueξi(for i=1,...,s),whereξdenotes the corresponding primitive root of unity.Thenσcan be extended to aﬁnite-order automorphism of the simple vertex operator algebra L g(k,0)which admits the following decompositionL g(k,0)=L g(k,0)0⊕L g(k,0)1⊕···⊕L g(k,0)swhereL g(k,0)i={v∈L g(k,0)|σ(v)=ξi v}.Clearly L g(k,0)i is a g0–module.For a dominant weightµfor g0we deﬁne µ=k′Λ0+µ.Denote by L g(k′,µ)the irreducible highest weight g0–module with highest weight µ.8DRAˇZEN ADAMOVI´C AND OZREN PERˇSENote that the lowest conformal weight of any g0–module with highest weight µis given by the formula(µ,µ+2ρ0)09Table1.g,g0are simple Lie algebras,g0<g.g decomposition of g k′Cℓ−1Dℓg0⊕V g0(ω1)−ℓ+2F4−3G2g0⊕V g0(ω1)⊕V g(ω2)−5/3Dℓ−ℓ+3/210DRAˇZEN ADAMOVI´C AND OZREN PERˇSELemma4.3.We have:(1)The lowest conformal weights of A(1)2–modules with highest weights−113Λ0+Λ1+Λ2are54,respectively.(2)The lowest conformal weights of B(1)ℓ−1–modules with highest weights−ℓΛ0+ 2Λ1and−ℓΛ0+Λ2are2+1ℓ−1,respectively.(3)The lowest conformal weights of D(1)ℓ–modules with highest weights(−ℓ−12)Λ0+Λ2are2+22ℓ−1,respectively.Lemma4.4.As an F(1)4–module,L E6(−3,0)does not contain singular vec-tors of weights−7Λ0+2Λ4,−5Λ0+Λ1and−7Λ0+Λ3.Proof.Weﬁrst note that F(1)4–modules with highest weights−7Λ0+2Λ4and−5Λ0+Λ1do not have integral conformal weights.(The correspondinglowest conformal weights are132,respectively.)The lowest conformal weight of module of highest weight−7Λ0+Λ3forF(1) 4is2,but one can directly check that there is no non-trivial singular vec-tor of that weight in L E6(−3,0).We use the construction of the root systemof type E6from[11],[23].For a subset S={i1,...,i k}⊆{1,2,3,4,5}withodd number of elements(i.e k=1,3or5),denote by e(i1...i k)the(suitablychosen)root vector associated to positive root1115.Conformal embedding of L G 2(−2,0)into L B 3(−2,0)andL D 4(−2,0)In this section we consider the conformal embedding associated to a pair of simple Lie algebras (D 4,G 2).It turns out that this case is more complicated and one can not directly apply general results from Section 3.The main diﬀerence between this case and cases studied in Section 4is that L D 4(−2,0)contains a non-trivial singular vector for G (1)2of conformal weight 2.In order to prove that L D 4(−2,0)is completely reducible L G 2(−2,0)–module we shall need more precise analysis,which uses the conformal embedding of L B 3(−2,0)in L D 4(−2,0)from Section 4.Let g be the simple complex Lie algebra of type D 4.Then g has an order three automorphism θ,induced from the Dynkin diagram automorphism,such that the ﬁxed point subalgebra g 0is isomorphic to the simple Lie algebra of type G 2.We have the decomposition of g into eigenspaces of θ:g =g 0⊕V (1)g 0(ω1)⊕V (2)g 0(ω1),(5.1)where V (1)g 0(ω1)is generated by highest weight vector e ǫ1−ǫ4+ξe ǫ1+ǫ4+ξ2e ǫ2+ǫ3and V (2)g 0(ω1)by e ǫ1−ǫ4+ξe ǫ2+ǫ3+ξ2e ǫ1+ǫ4,for suitably chosen root vectors.(Here ξdenotes the primitive third root of unity.)Lie algebra g 0is also a subalgebra of Lie algebra of type B 3considered in Section 4,which we now denote by g ′.We have g ′=g 0⊕V (3)g 0(ω1),(5.2)where V (3)g 0(ω1)is generated by highest weight vector e ǫ1−ǫ4+e ǫ1+ǫ4−2e ǫ2+ǫ3.By using (5.2),the decomposition of g from Section 4and the fact that V B 3(ω1)∼=V G 2(ω1)as g 0–module,one obtains another decomposition of g :g =g 0⊕V (3)g 0(ω1)⊕V (4)g 0(ω1),(5.3)where V (4)g 0(ω1)is generated by e ǫ1−ǫ4−e ǫ1+ǫ4.By using (5.2)and results from Section 2we see that L G 2(−2,0)is conformally embedded into L B 3(−2,0),which is conformally embedded in L D 4(−2,0).Recall that Theorem 4.1gives that the vertex operator algebra L D 4(−2,0)has an order two automorphism σwhich deﬁnes the following decomposition:L D 4(−2,0)=L B 3(−2,0)⊕L B 3(−2,ω1).(5.4)Denote by L (i )G 2(−2,ω1)the G (1)2–submodule of L D 4(−2,0)generated by top component V (i )g 0(ω1),for i =1,2,3,4.Furthermore,one can directly check that vectorv =(e ǫ1−ǫ3(−1)e ǫ1−ǫ4(−1)+e ǫ2−ǫ4(−1)e ǫ2+ǫ3(−1)+e ǫ2+ǫ4(−1)e ǫ1+ǫ4(−1)−e ǫ1−ǫ3(−1)e ǫ1+ǫ4(−1)−e ǫ2−ǫ4(−1)e ǫ1−ǫ4(−1)−e ǫ2+ǫ4(−1)e ǫ2+ǫ3(−1))112DRA ˇZEN ADAMOVI ´CAND OZREN PER ˇSE is the unique (up to a scalar)non-trivial singular vector of weight −5Λ0+Λ2for G (1)2in L D 4(−2,0).Denote by L G 2(−2,ω2)the G (1)2–submodule gener-ated by v .We also have that θ(v )=v and σ(v )=−v .This implies that v ∈L B 3(−2,ω1).Now we consider L B 3(−2,0)and L B 3(−2,ω1)as G (1)2–modules.We use the following lemmas:Lemma 5.1.We have the following tensor product decompositions:(1)V G 2(ω1)⊗V G 2(ω1)=V G 2(2ω1)⊕V G 2(ω2)⊕V G 2(ω1)⊕V G 2(0),(2)V G 2(ω1)⊗V G 2(ω2)=V G 2(ω1+ω2)⊕V G 2(2ω1)⊕V G 2(ω1).Lemma 5.2.As a G (1)2–module,L D 4(−2,0)does not contain singular vec-tors of weights −6Λ0+2Λ1and −7Λ0+Λ1+Λ2.Proof.The proof follows from the observation that G (1)2–modules of highest weights −6Λ0+2Λ1and −7Λ0+Λ1+Λ2do not have integral conformal weights.(The corresponding lowest conformal weights are 72,respec-tively.) Proposition 5.3.We have:(1)L B 3(−2,0)= L G 2(−2,0)+ L (3)G 2(−2,ω1).(2)L B 3(−2,ω1)= L(4)G 2(−2,ω1)+ L G 2(−2,ω2).Proof.(1)Since v is the unique (up to a scalar)singular vector of weight −5Λ0+Λ2for G (1)2in L D 4(−2,0),it follows that there is no singular vec-tor of that weight in L B 3(−2,0).The tensor product decomposition from Lemma 5.1(1),Lemma 5.2and standard fusion rules arguments now im-ply that L G 2(−2,0)+ L (3)G 2(−2,ω1)is a vertex subalgebra of L B 3(−2,0)which contains generators of L B 3(−2,0).The claim (1)follows.Claim (1),Lemmas 5.1and 5.2and standard fusion rules arguments now implythat L (4)G 2(−2,ω1)+ L G 2(−2,ω2)is L B 3(−2,0)–submodule of the irreducible module L B 3(−2,ω1).This implies claim (2). Theorem 5.4.We have:L D 4(−2,0)=L G 2(−2,0)⊕L G 2(−2,ω2)⊕L (1)G 2(−2,ω1)⊕L (2)G 2(−2,ω1).Proof.First we notice that relation (5.4)and Proposition 5.3give L D 4(−2,0)= L G 2(−2,0)+ L G 2(−2,ω2)+ L (3)G 2(−2,ω1)+ L (4)G 2(−2,ω1),which also impliesL D 4(−2,0)= L G 2(−2,0)⊕ L G 2(−2,ω2)⊕ L (1)G 2(−2,ω1)⊕ L (2)G 2(−2,ω1).By using the fact that Z 3–orbifold components of simple vertex operator algebra are simple (cf.[14]),we get thatV = L G 2(−2,0)⊕ L G 2(−2,ω2)13 is a simple vertex operator algebra,and that L(1)G2(−2,ω1)and L(2)G2(−2,ω1) are its irreducible modules.Since the automorphismσacts as−1on L G2(−2,ω2),it follows that V is Z2–graded,so its graded components are also simple.Theorem3from[14]now implies that L(1)G2(−2,ω1)and L(2)G2(−2,ω1)are inequivalent V–modules,and Theorem6.1from the same paper then implies that L(1)G2(−2,ω1)and L(2)G2(−2,ω1)are irreducible asL G2(−2,0)–modules.The proof follows. It follows from the proof of Theorem5.4that L(3)G2(−2,ω1)and L(4)G2(−2,ω1)are also irreducible L G2(−2,0)–modules,so we obtainCorollary5.5.We have:(1)L B3(−2,0)=L G2(−2,0)⊕L G2(−2,ω1),(2)L B3(−2,ω1)=L G2(−2,ω1)⊕L G2(−2,ω2).References[1] D.Adamovi´c,Some rational vertex algebras,Glas.Mat.Ser.III29(49)(1994),25-40.[2] D.Adamovi´c,A construction of some ideals in aﬃne vertex algebras,Int.J.Math.Math.Sci.(2003),971–980.[3] D.Adamovi´c and as,Vertex operator algebras associated to modular in-variant representations for A(1)1,Math.Res.Lett.2(1995),563–575.[4] D.Adamovi´c and O.Perˇs e,On coset vertex algebras with central charge1,Math.Commun.15(2010),143–157.[5] D.Adamovi´c and O.Perˇs e,The vertex algebra M(1)+and certain aﬃne vertexalgebras of level−1,arXiv:1006.1752.[6]R.C.Arcuri,J.F.Gomes and D.I.Olive,Conformal subalgebras and symmetricspaces,Nuclear Phys.B285(1987),327–339.[7]J.D.Axtell and K.H.Lee,Vertex Operator Algebras Associated to Type GAﬃne Lie Algebras,arXiv:1011.3473v1.[8] F.A.Bais and P.G.Bouwknegt,A classiﬁcation of subgroup truncations of thebosonic string,Nuclear Phys.B279(1987),561–570.[9]J.Bockenhauer,D.E.Evans and Y.Kawahigashi,Chiral structure of modularinvariants for subfactors,Comm.Math.Phys.210(2000),733–784.[10]R.E.Borcherds,Vertex algebras,Kac-Moody algebras,and the Monster,Proc.A83(1986),3068–3071.[11]Bourbaki,Groupes et alg`e bras de Lie,Hermann,Paris,1975.[12]P.Cellini,V.G.Kac,P.M¨o seneder Frajria and P.Papi,Decomposition rulesfor conformal pairs associated to symmetric spaces and abelian subalgebras of Z2-graded Lie algebras,Adv.Math.207(2006),156–204.[13] C.Dong,H.Li and G.Mason,Simple currents and extensions of vertex operatoralgebras,Comm.Math.Phys.180(1996)671–707.[14] C.Dong and G.Mason,On quantum Galois theory,Duke Math.J.86(1997),305–321.[15] D.E.Evans and T.Gannon,Modular invariants and twisted equivariant K-theory,Commun.Number Theory Phys.3(2009),209–296.[16] A.Felikson,A.Retakh and P.Tumarkin,Regular subalgebras of aﬃne Kac-Moody algebras,J.Phys.A41(2008),16pp.14DRAˇZEN ADAMOVI´C AND OZREN PERˇSE[17] E.Frenkel and D.Ben-Zvi,Vertex algebras and algebraic curves,MathematicalSurveys and Monographs,88,American Mathematical Society,Providence,RI, 2001.[18]I.Frenkel,Y.-Z.Huang and J.Lepowsky,On axiomatic approaches to vertexoperator algebras and modules,Mem.Amer.Math.Soc.104,1993.[19]I.Frenkel,J.Lepowsky and A.Meurman,Vertex Operator Algebras and the Mon-ster,Pure and Appl.Math.,Vol.134,Academic Press,Boston,1988.[20]I.Frenkel and Y.-C.Zhu,Vertex operator algebras associated to representationsof aﬃne and Virasoro algebras,Duke Math.J.66(1992),123–168.[21]J.Frohlich,J.Fuchs,I.Runkel and C.Schweigert,Algebras in tensor categoriesand coset conformalﬁeld theories,Fortsch.Phys.52(2004),672–677.[22]J.Fuchs,Aﬃne Lie algebras and quantum groups.An introduction with applica-tions in conformalﬁeld theory,Cambridge Monographs on Mathematical Physics.Cambridge:Cambridge University Press.xiv,433p.(1992).[23]J.E.Humphreys,Introduction to Lie algebras and representation theory,Grad-uate Texts in Mathematics,Vol.9.Springer-Verlag,New York-Berlin,1972.xii+169pp.[24]V.G.Kac,Inﬁnite dimensional Lie algebras,3rd ed.,Cambridge Univ.Press,Cambridge,1990.[25]V.G.Kac,Vertex Algebras for Beginners,University Lecture Series,SecondEdition,AMS,Vol.10(1998).[26]V.Kac and M.Wakimoto,Modular invariant representations of inﬁnite dimen-sional Lie algebras and superalgebras,A85(1988), 4956–4960.[27]V.Kac and M.Wakimoto,Classiﬁcation of modular invariant representations ofaﬃne algebras,in Inﬁnite Dimensional Lie algebras and groups,Advanced Series in Math.Phys.7,World Scientiﬁc,Teaneck NJ,1989.[28] A.Kirillov and V.Ostrik,On a q-analogue of the McKay correspondence andthe ADE classiﬁcation of sl2conformalﬁeld theories,Adv.in Math.171(2002), 183–227.[29]J.Lepowsky and H.Li,Introduction to vertex operator algebras and their repre-sentations,Progress in Math.,Vol.227,Birkhauser,Boston,2004.[30]H.-S.Li,Local systems of vertex operators,vertex superalgebras and modules,J.Pure Appl.Algebra109(1996),143–195.[31]H.Li,Extension of vertex operator algebras by a self-dual simple module,J.Algebra187(1997),236–267.[32]O.Perˇs e,Vertex operator algebras associated to type B aﬃne Lie algebras onadmissible half-integer levels,J.Algebra307(2007),215–248.[33]O.Perˇs e,Vertex operator algebra analogue of embedding of B4into F4,J.PureAppl.Algebra211(2007),702–720.[34]O.Perˇs e,Vertex operator algebras associated to certain admissible modules foraﬃne Lie algebras of type A,Glas.Mat.Ser.III43(63)(2008),41–57. [35]O.Perˇs e,Embeddings of vertex operator algebras associated to orthogonal aﬃneLie algebras,Comm.Algebra38(2010),1761–1778.[36] A.N.Schellekens and N.P.Warner,Conformal subalgebras of Kac-Moody alge-bras,Phys.Rev.D34(1986),3092–3096.[37]M.Wakimoto,Lectures on inﬁnite-dimensional Lie algebra,World Scientiﬁc,River Edge NJ,2001.F aculty of Science-Department of Mathematics,University of Zagreb, CroatiaE-mail address:adamovic@math.hr;perse@math.hr。

On the reflexivity of point sets

FACHBEREICH3MATHEMATIKO N THE R EFLEXIVITY OF P OINT S ETSbyE STHER M.A RKIN S´ANDOR P.F EKETEF ERRAN H URTADO J OSEPH S.B.M ITCHELLM ARC N OY V ERA S ACRIST´AN S AURABH S ETHIANo.694/2000On the Reﬂexivity of Point SetsEsther M.Arkin S´a ndor P.Fekete Ferran Hurtado Joseph S.B.MitchellMarc Noy Vera Sacrist´a n Saurabh SethiaAbstractWe introduce a new measure for planar point sets.Intuitively,it describes the combinatorial distance from a convex set:The reﬂexivity of is given by the smallest number of reﬂex vertices ina simple polygonization of.We prove various combinatorial bounds and provide efﬁcient algorithms tocompute reﬂexivity,both exactly(in special cases)and approximately(in general).Our study naturallytakes us into the examination of some closely related quantities,such as the convex cover numberof a planar point set,which is the smallest number of convex chains that cover,and the convex partitionnumber,which is given by the smallest number of disjoint convex chains that cover.Weprove that it is NP-complete to determine the convex cover or the convex partition number,and we givelogarithmic-approximation algorithms for determining each.1IntroductionIn this paper,we study a fundamental combinatorial property of a discrete set,,of points in the plane: What is the minimum number,,of reﬂex vertices among all of the simple polygonalizations of?A polygonalization of is a closed tour on whose straight-line embedding in the plane deﬁnes a connected cycle without crossings,i.e.,a simple polygon.A vertex of a simple polygon is reﬂex if it has interior angle greater than.We refer to as the reﬂexivity of.In general,there are many different polygonalizations of a point set,.There is always at least one:sim-ply connect the points in angular order about the center of mass.A set has precisely one polygonization, iff it is in convex position,but usually there is a great number of them.Studying the set of polygonizations (e.g.,counting them,enumerating them,or generating a random element)is a challenging active area of investigation in computational geometry.The reﬂexivity quantiﬁes,in a combinatorial sense,the degree to which the set of points is in convex position.See Figure1for an example.We conduct a formal study of reﬂexivity,both in terms of its combinatorial properties and in terms of an algorithmic analysis of the complexity of computing it,exactly or approximately.Some of our attention is focussed on the closely related convex cover number of,which gives the minimum number of convex chains(subsets of in convex position)that are required to cover all points of.For this question,we distinguish two cases:The convex cover number,,is the smallest number of convex chains to cover; the convex partition number,,is the smallest number of convex chains with pairwise-disjoint convex hulls to cover.(Note that nested chains are feasible for a convex cover,but not for a convex partition.)Figure1:Two polygonalizations of a point set,one(left)using7reﬂex vertices and one(right)using only3 reﬂex vertices.Motivation.In addition to the fundamental nature of the questions and problems we address,we are also motivated to study reﬂexivity for several other reasons:(1).An application motivating our original investigation is that of meshes of low stabbing number and their use in performing ray shooting efﬁciently.If a point set has low reﬂexivity or convex partition number,then it has a triangulation of low stabbing number,which is much lower than the generala problem that has also been studied by Aggarwal et al.[2].We will see in Section4that there are further connections.The number of polygonizations on points is in general exponential in and to give tight bounds on the maximum value attainable for a given has also been object of intensive research([11]).The minimum number of reﬂex vertices among all the polygonizations of a point set is the reﬂexivity of,a concept we introduce in this work.Main Results.The main results of this work include:Tight combinatorial bounds for the worst-case reﬂexivity in a number of cases,including the general case and the case of onion depth2.Upper and lower combinatorial bounds for reﬂexivity,convex cover number,convex partition number, and for their relative behavior.We obtain exact worst-case values for small cardinalities.Proofs of NP-completeness for computing convex cover and convex partition numbers.Algorithmic results yielding approximations for convex cover number,convex partitioning number,and(Steiner)reﬂexivity.We also give efﬁcient exact algorithms for cases of low reﬂexivity. 2PreliminariesThroughout this paper,will be a set of points in the plane.Let be any polygonization of.We say that is simple if edges may only share common endpoints,and each endpoint is incident to exactly two edges.Let be the set of all polygonizations of.Note that is not empty,since any point set having points has at least one polygonalization(e.g.,the star-shaped polygonalization obtained by sorting points of angularly about a point interior to the convex hull of).A simple polygon is a closed Jordan curve,subdividing the plane into an unbounded and a bounded component.We say that the bounded component is the interior of.A reﬂex vertex of a simple polygon is a common endpoint of two edges,such that the interior angle between these edges is larger than.We say that an angle is convex if it is not reﬂex.We deﬁne to be the number of reﬂex vertices in, and to be the number of non-reﬂex,i.e.,convex vertices in.We deﬁne the reﬂexivity of a planar point set to be.Similarly,the convexity of a planar point set is deﬁned to be .Note that.The convex hull of a set is the smallest convex set that contains all elements of;the convex hull elements of are the members of that lie on the boundary of the convex hull.The layers of a point set are given by repeatedly removing all convex hull elements,and considering the convex hull of the remaining set.We say that has layers or onion depth if this process terminates after precisely layers.A set forms a convex chain(or is in convex position)if it has only one layer.A Steiner point is a point not in the set that may be added to in order to improve some structure of.We deﬁne the Steiner reﬂexivity to be the minimum number of reﬂex vertices of any simple polygon with vertex set.Similarly, we can deﬁne the Steiner convexity.(Furthermore,Steiner points can be required to lie within the convex hull,or be arbitrary.In this abstract,we do not elaborate on this difference.)In the following,we use the notation and for the worst-case values for sets of size.For a givenﬁnite set,let be the family of all sets of convex chains,such that each element of is part of at least one chain.(For obvious reasons,we say that a set of chains is called a convex cover of.)Similarly,is the family of all convex covers of for which the convex hulls of any two chains are mutually disjoint.Then we deﬁne the convex cover number as the smallest size of a convex cover of,and the convex partition number .Again,we denote by and the worst-case values for sets of size.Finally,we state a basic property of polygonalizations of point sets.The proof is straightforward. Lemma2.1In any polygonization of,the points on the convex hull of are always convex vertices,and they occur in the polygonization in the same order in which they occur along the convex hull.33Combinatorial BoundsIn this section we establish several combinatorial results on reﬂexivity and convex cover numbers.One of our main combinatorial results establishes an upper bound on the reﬂexivity of that is tight interms of the number of points interior to the convex hull of.Given that the points of that are verticesof the convex hull are required to be convex vertices in any(non-Steiner)polygonalization of,the boundin terms of seems to be quite natural.Theorem3.1Let be a set of points in the plane,of which are interior to the convex hull.Then,.Figure2:Computing a polygonalization with at most reﬂex vertices.Proof:We describe a polygonization in which at most half of the interior points are reﬂex.We begin with thepolygonalization of the convex hull vertices that is given by the convex polygon bounding the hull.We theniteratively incorporate other(interior)points of into the polygonalization.Fix a point that lies on theconvex hull of.At a generic step of the algorithm,the following invariants hold:(1)our polygonalizationconsists of a simple polygon,,whose vertices form a subset of;and(2)all points of that are not vertices of lie interior to;in fact,the points all lie within the subpolygon,,to the left ofthe diagonal,where is a vertex of such that the subchain of from to(counter-clockwise)together with the diagonal forms a convex polygon().Deﬁne to be theﬁrst point of thatis encountered when sweeping the ray counter-clockwise about its endpoint.Then,we sweep thesubray with endpoint further clockwise,about,until we encounter another point,,of.(If ,we can readily incorporate into the polygonalization,increasing the number of reﬂex vertices by one.)Now,the ray intersects the boundary of at some point on the boundary of.We now modify to include interior points and(and possibly others as well)by replacing theedge with the chain,where the points are interior points that occur along the chain we obtain by“pulling taut”the chain(i.e.,by continuing,in the“gift wrapping”fashion,to rotate rays counter-clockwise about each interior point that is hit until we encounter).In this way we incorporate at least two new interior points(of)into the polygonalization,while creating only one new reﬂex vertex(at).It is easy to check that the invariants(1)and(2)hold after this step.In fact,the upper bound of Theorem3.1,,is tight,as we now argue based on the special conﬁguration of points,,in Figure3.The set is deﬁned for any integer,as follows: points are placed in convex position(e.g.,forming a regular-gon),forming the convex hull ,and the remaining interior points are also placed in convex position,each one placed4“just inside”,near the midpoint of an edge of.The resulting conﬁguration has two layers in its convex hull.Lemma3.1,below,shows that.Figure3:Left:The conﬁguration of points,,which has reﬂexivity.Right:A polygonalization having reﬂex vertices.Lemma3.1For any,.Proof:Denote by the points on the convex hull,,and the points“just inside”the convex hull,,where is along the convex hull edge.From Lemma2.1we know that points are connected in their order around the convex hull,and are all convex vertices in any polygonization.Consider an arbitrary pocket of a polygonization,having lid and let denote the number of interior points that go to this pocket(if the convex hull edgebelongs to the polygonization,with a slight abuse of notation we can consider it as a pocket with ).Observe that implies that belongs to this particular pocket.If the pocket contains a single interior point,namely,and then is a reﬂex vertex in this polygonization.To complete our proof we will show that if this pocket contains more interior points,among them only will be a convex point.We use the following simple fact:Given a set of points,all but one of which is in convex position,all polygonizations of this set have a unique reﬂex vertex,namely the point not on the convex hull.The pocket with lid includes points,;if is not the only interior point included in this pocket,then this pocket together with the lid is a polygon as in the simple fact.Therefore the polygon which is the pocket has only one reﬂex vertex,,and when considered“inside-out”as a pocket of the original polygon,only among the interior points is a convex vertex.Therefore the number of reﬂex vertices in a pocket is in any case at least,and we haveRemark.Since,the corollary below is immediate from the theorem.The gap in the bounds for ,between and,remains an intriguing open problem.While our combinatorial bounds are tight in terms of(the number of points of whose convexity/reﬂexivity is not forced by the convex hull of),they are not yet tight in terms of.Corollary3.1.Steiner Points.If we allow Steiner points in the polygonalizations of,the reﬂexivity of may go down, as the example in Figure4shows.At this point,it is unclear whether this estimate characterizes a worst case.We believe it does: Conjecture1For a set of points in the plane,we have.In terms of the cardinality of,we obtain the following combinatorial bounds:5Figure4:Above:A point set having reﬂexivity.Below:The reﬂexivity of when Steiner points are permitted is substantially reduced from the no-Steiner case:.Theorem3.2For a set of points in the plane,we have.Proof:Sort the points by their-coordinates and group them into consecutive triples.Consider point with a very large positive-coordinate and point with a very negative-coordinate.Each triple together with either point or point forms a convex quadrilateral.Connecting up the convex quadrilaterals,by making perturbed copies of points and,while adding another hull Steiner point on the left,yields a polygonization with at most reﬂex points,as shown in Figure5.Figure5:Polygonalization of points using only reﬂex(Steiner)points.By a careful analysis of our example in Figure3,we can show the following:Theorem3.3If one allows Steiner points that are interior to,then any Steiner polygonalization of has at least reﬂex vertices.Two-Layer Point Sets.Let be a point set that has onion depth2.It is clear from our repeated use of the example in Figure3that this is a natural case that is a likely candidate for worst-case behavior.With a very careful analysis of this case,we are able to obtain tight combinatorial bounds on the reﬂexivity in terms of :Theorem3.4Let be a set of points having two layers.Then,and this bound is tight.6Proof:Fact1In any polygonization of a set of points,all pockets have at least one reﬂex vertex.Proof:This follows from the fact that the polygon deﬁned by the pocket and its convex hull edge must have at least three convex vertices(as does any simple polygon).Consider a set of points with onion depth two.Let be the number of points on the convex hull. Thus there are points on the interior onion layer.Let the points on the convex hull bein clockwise order.Let the points on the interior onion layer be in clockwise order.All arithmetic involving the subscripts of the’s and’s is done mod and mod,respectively.Consider any point on the interior onion layer.Let the intersection of ray with the convex hull be.We call the directed segment the spoke originating at,and we say that the spoke belongs to the convex hull segment and the segment has the spoke.Refer to Figure6.The number of spokes a convex hull segment has is called its spoke count.Let be the spoke count of convex hull segment.Clearly there are spokes and each spoke belongs to exactly one convex hull segment(assuming no degeneracy).Also for all,and .Figure6:The situation for a point set with two layers.Consider any convex hull edge with non-zero spoke count.Assume that the originating points of its spokes are,,,in clockwise order.Then the pocket,,,,,is called the standard pocket for the convex hull edge.See Figure7.Figure7:Standard pockets.Fact2A standard pocket has exactly one reﬂex vertex.7Proof:Vertices are convex,since the angles at these vertices are the interior angles of the inner onion layer.The vertex is convex because point is on the right of directed line .The vertex is reﬂex because there has to be at least one reﬂex vertex in any pocket. Fact3No two standard pockets intersect each other,except possibly at their endpoints.Proof:The annulus between the two onion rings is divided into disjoint regions by the spokes.A standard pocket for segment may share a region with each of the standard pockets(if any)for segments and.In such shared regions two pockets have a segment each.The two segments do not intersect because the two segments are obtained by rotating two spokes about their points of origin until their intersection with the convex hull reaches a vertex.The two segments can hence either remain disjoint or share an endpoint.We obtain the standard polygonalization by connecting all standard pockets,in order,using convex hull segments with zero spoke count.See Figure8.Fact4The number of reﬂex vertices in the standard polygonization is at most.Proof:The number of standard pockets can neither exceed the number of convex hull segments nor the number of internal points,.Hence,this polygonization has at most standard pockets and hence reﬂex vertices.This can be at most.Figure8:Standard polygonalization.Consider a convex hull segment that has a non-zero spoke count.Assume that the origins of its spokes are.We call the pocket,,,,,the premium pocket for convex hull segment.Note that the premium pocket of a segment has one more point than its standard pocket.We obtain the premium polygonization as follows:Start with the convex hull.Process convex hull segments in clockwise order beginning anywhere.If the convex hull segment has spoke count greater than or equal to two,replace the edge with its standard pocket.If the spoke count is zero,do nothing.If the spoke count is one,move to the next segment with non-zero spoke count and replace it with its premium pocket provided that it is not already processed.If the next segment with non-zero spoke count was already processed,replace the segment being processed with its standard pocket,and we are done.One can verify that this process gives a valid polygonalization.Fact5The number of reﬂex vertices in a premium polygonization is at most.Proof:In the premium polygonization,each pocket has at least one reﬂex and one convex vertex,with the exception of the last pocket created,which may have only a single reﬂex vertex.Thus,the number of pockets created cannot exceed.Also,the number of pockets cannot exceed the number of convex hull edges,.Thus,the number of pockets and,therefore,the number of reﬂex vertices cannot8exceed,which can be at most.We now deﬁne the intruding polygonization,as follows:Start with the convex hull.Process convex hull segments in clockwise order,beginning anywhere.We consider cases,depending on the spoke count,,of the convex hull segment:Do nothing.Replace with a standard pocket.Replace the next non-zero count unprocessed hull segment with its premium pocket.(If no such non-zero count unprocessed segment exists,then replace by its standard pocket and stop.)We distinguish two subcases:Case A If this is the last segment to be processed or if the sequence of spoke counts following this segment begins with either0,or a count,or an odd number of1’s,or an even number of1’sfollowed by a zero,do the following:Replace with its standard pocket.Case B If there are an even number of1’s followed by a non-zero spoke count,or if the sequence begins with2or3,do one of the following,depending on the current“mode”;initially,the modeis“normal.”Normal Mode.Replace with its premium pocket.This leaves one of the originatingpoints of;call a point of this type the pending point.Go into“Point Pending Mode.”Point Pending Mode.Let the origins of the spokes of be.Find theﬁrstpoint of intersection of the ray with the current polygonization.There are only two choicesfor where the point of intersection can lie.If the point of intersection lies on,treat ray as a spoke of,thus increasingby1.Now replace by its standard pocket,and return to the“Normal Mode.”If the point of intersection lies on,replace with.This reduces from2to1.Now handle as in Case1.Fact6The intruding polygonalization produces a valid(non-crossing)polygonalization of,with at most reﬂex vertices.Proof:It is straightforward to check each case to make certain that the polygonalization is valid;see the full paper.We prove the bound on the number of reﬂex vertices by charging each reﬂex vertex created to a set of4input points.In case,we do not create any reﬂex points.In case,we create a reﬂex vertex as part of the standard pocket.We charge this to the three internal vertices of the pocket and the source vertex of the segment being processed.In case,we create a reﬂex vertex as part of the premium pocket of the next segment having non-zero spoke count.We charge this to the sources of the segment being processed and the next segment and to the(at least two)internal vertices of the pocket.If and we are in Case A,we create a reﬂex vertex as part of the standard pocket.We charge this to the source of the segment being processed and the two internal vertices.We need to charge it to one more point.If the next segment has zero spoke count,we charge it to its source.If the next segment has spoke count greater than or equal to four,we charge it to one of its internal vertices.(Recall that for a segment with spoke count greater than or equal to four,only three of its internal vertices will get charged by its own standard pocket.)9If there are some number of1’s followed by a0,we charge the source of segment having spoke count 0.If there are odd number of1’s followed by a non-zero count,then note that the segment following this sequence of1’s will be replaced by its premium pocket by our algorithm.This segment will have a spoke count greater than or equal to2.We put the extra charge needed on one of the internal points of this premium pocket.(Recall that this premium pocket will have at least3internal points and the pocket itself will charge only two of them.)If and we are in Case B,we consider each of the two modes separately:Normal Mode If the sequence begins with a2or3,we charge the reﬂex vertex created to the internal points (at least3)of the premium pocket and the source of segment being replaced.If the sequence has an even number of1’s followed by a non-zero spoke count,then we have a case similar to that above,but we have only two internal points to charge in the premium pocket.However, since theﬁrst segment with spoke count1will be replaced by its premium pocket,there will be an odd number of1’s remaining.Hence,the segment following this sequence of one(which has spoke count)will be replaced with a premium pocket,which has an extra internal point to which we can charge.Note that in Normal Mode,in both of the cases above,the source of the segment being processed remains uncharged.Also,the pending point remains to be incorporated into the polygonization. Point Pending Mode Depending on where ray intersects in this case,there are two possibilities: If the point of intersection lies on,then we are creating a pocket with at least2internal points.We charge the reﬂex vertex created to the two internal points and to the source of the segment being processed and to the uncharged point in the previous Normal Mode.See Figure9.Figure9:Subcase of Case B,:Ray intersects segment,but not segment.If the point of intersection lies on,then the subpocket has two internal points.In this case we charge its reﬂex point(which is the pending point)to these two internal points and to the uncharged point in the previous normal mode.We need to make one more charge,which depends on the spoke count of next segment.If it is2or3,we charge the extra internal point in its premium pocket.If it is1,note that since there were even number of1’s following,after we replace the next one with its premium pocket,there would be odd number of1’s remaining.Hence,the segment following this10Figure10:Subcase of Case B,:Ray intersects segment,but not segment.sequence of1’s(which has non-zero spoke count),will be replaced with its premium pocket,by our algorithm.Hence,we can charge its extra internal point.See Figure10.This concludes the proof of Theorem3.4Furthermore,we can prove the following lower bound for polygonizations with a very special structure that may be useful in an inductive proof in more general cases:Lemma3.2For a point set with onion depth a polygonization with at most301412512622722823933Table1:Worst-case values of,,for small values of.Theorem3.7For a planar set,we have the estimates,and these upper bounds are best possible.Proof:The upper bound for by is trivial.The upper bound for by can be found in[4].Figure11:An example with.(Thick lines indicate the presence of many points.) The example in Figure11yields a class of examples with,which is the worst example we know of.On the other hand,it is a surprisingly difﬁcult open problem to prove that there is some bounded ratio between and:Conjecture2For a set of points in the plane,we have.However,it is not hard to see that the estimate holds(see the proof of Corollary5.1), so a proof of Conjecture2would follow from the validity of Conjecture1.Small Point Sets.It is natural to consider the exact values of,,and for small values of .Table1below shows some of these values.It is not really difﬁcult to push these bounds a little further, but the computational effort becomes increasingly tedious.4ComplexityTheorem4.1It is NP-complete to decide whether for a planar point set the convex partition number is below some threshold.Proof:We give a reduction from P LANAR3S AT,which was shown to be NP-complete by Lichtenstein (see[17]).(A3S AT instance is called a P LANAR3S AT instance,if the(bipartite)“occurrence graph”is planar,where consists of a vertex or per variable or clause,and two vertices are adjacent if represents a variable that occurs in the clause represented by.See Fig-ure12(a)for an example,where a solid edge denotes an unnegated literal,while a dashed edge represents a negated literal in a clause.The basic idea is the following:Each variable is represented by a set of points that can be partitioned into disjoint convex chains in two different ways.One of these possibilities will correspond to a setting12of“true”,the other to a setting of“false”.Each clause is represented by a set of points,such that it can be covered by three convex chains disjoint from all other chains,iff at least one of the variables is set in a way that satisﬁes the clause.So let be a P LANAR3S AT instance with variables and clauses.Consider a straight-line embedding of its occurrence graph.In aﬁrst step,for every vertex representing a variable,draw a small polygon with edges around,where is bounded by the maximum degree of a variable vertex.(See Figure12(b).)This is done such that no edge of passes through one of the corners of, and only edges adjacent to intersect;moreover,we choose edge orientations for all polygons that assure that no two different polygons and have two collinear edges(See Figure12(c),where we have and used rectangles as polygons to keep theﬁgure clear.)Furthermore,we choose the polygons such that all the line segments connecting to clauses where appears unnegated intersect“even”edges of the polygon,and all the line segments connecting to clauses where appears negated intersect“odd”edges of the polygon.In a second step,replace the polygons(and thus the variable vertices)by appropriate sets ofpoints–see Figure12for an example.Each of the corners of a polygon is represented by a dense convex chain of points,forming a convex curve with an opening of roughly,and an additional“pivot”point.Finally,we describe how to represent the clauses:Each vertex in representing a clause is replaced by three points forming a small triangle;furthermore,for each of the three edges connecting some variable vertex to,add a convex chain of points within the polygon.is lined up with the triangle such that is a convex chain.On the other hand,the opening of towards is narrow enough to prevent any other point from forming a convex chain with all points in.We also avoid any other collinearities in the overall arrangement.Now we claim the following correspondence:The resulting point set can be partitioned into at most disjoint convex chains,iff the P LANAR 3S AT instance is satisﬁable.To see that a satisfying truth assignment induces a feasible decomposition,note that there are exactly two ways to cover the points for a polygon with edges by at most disjoint convex chains.(One of these choices arises by joining the pairs of sets that belong to the“even”edges of a polygons box into one convex chain,the other by joining the pairs of sets for the“odd”edges of a polygon.This is shown in Figure12(d).) For a true variable,take the“even”choice,for a false variable,the“odd”choice.Now consider a clause and a variable that satisﬁes.By the choice of chains covering the points for,we join the triangle for with the chain into one convex chain,without intersecting any other chain.The other two chains are each covered by separate chains.This yields a decomposition into disjoint convex chains, as claimed.To see the converse,assume we have a decomposition into at most disjoint convex chains.We start by considering how the sets and(henceforth called“gadget sets”,with )can be covered in such a solution.Associate each chain with all the of which it covers at least points.It is straightforward to see that a convex chain that covers at least points from each of three different gadget sets must contain some other point of the set in its interior,so this cannot occur in the given feasible decomposition.Therefore,no chain can be associated with more than two gadget sets.Moreover,a convex chain can contain at least points from each of two different gadget sets without any pivot points in its interior,only if these sets are some and.(In particular,it is not hard to see that no chain that covers points of a set can cover points from any other gadget set.)Since there are chains in total,there must be at least one chain associated with each gadget set,and a chain associated with a set cannot be associated with any other set.This means that13。

信息茧房用于什么作文主题

信息茧房用于什么作文主题英文回答：In today's hyper-connected world, individuals are constantly exposed to a vast array of information. While this access to knowledge has undoubtedly brought about many benefits, it has also given rise to a phenomenon known as the echo chamber or filter bubble. An echo chamber is aself-reinforcing environment in which individuals are only exposed to information that confirms their existing beliefs and opinions. This can lead to a narrow and distorted view of the world, as individuals become less likely to encounter opposing viewpoints or challenge their own assumptions.Echo chambers can manifest in various forms, but one of the most prevalent is through the use of social media and personalized news feeds. These platforms utilize algorithms that track user activity and preferences to deliver content that is tailored to their interests. While this can lead toa more personalized and enjoyable user experience, it also reinforces the echo chamber effect by limiting exposure to diverse perspectives.The consequences of living in an echo chamber can befar-reaching. By isolating individuals from opposing viewpoints, echo chambers can hinder critical thinking and make it more difficult to engage in meaningful dialoguewith those who hold different beliefs. This can lead to polarization and a lack of understanding between different segments of society.Furthermore, echo chambers can also have a detrimental impact on decision-making. When individuals are only exposed to information that supports their existing beliefs, they are less likely to consider alternative perspectivesor make informed choices. This can have significant implications for areas such as politics, public health, and scientific research.To mitigate the effects of echo chambers, it isessential to actively seek out diverse perspectives andchallenge our own assumptions. This can be done through reading a variety of news sources, engaging inconversations with individuals who hold different viewpoints, and actively seeking out information that contradicts our existing beliefs. By breaking out of our echo chambers, we can expand our understanding of the world, make more informed decisions, and foster a more inclusive and tolerant society.中文回答：信息茧房是指人们只接受和自己观点相符的信息，从而形成封闭的认知环境。

vertex centrality attributes -回复

vertex centrality attributes -回复What is Vertex Centrality?Vertex centrality is a concept used in network analysis to determine the importance or centrality of a particular vertex (or node) within a network. It measures the extent to which a vertex connects or interacts with other vertices in the network. Vertex centrality attributes, which are used to calculate vertex centrality, include degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality.1. Degree Centrality:Degree centrality measures the number of connections a vertex has within a network. It is calculated by counting the number of edges attached to a vertex. A vertex with a high degree centrality is considered important as it has a large number of direct connections or interactions with other vertices.2. Closeness Centrality:Closeness centrality measures the proximity of a vertex to all other vertices in the network. It quantifies how quickly information can spread from a vertex to other vertices in the network. Closenesscentrality is calculated by summing up the shortest path lengths between a vertex and all other vertices in the network. A vertex with a high closeness centrality is considered important as it can easily disseminate information to other vertices.3. Betweenness Centrality:Betweenness centrality measures the extent to which a vertex lies on the shortest paths between other pairs of vertices in the network. It quantifies the ability of a vertex to control the flow of information or resources between other vertices. Betweenness centrality is calculated by counting the number of times a vertex appears on the shortest path between every pair of vertices in the network. A vertex with a high betweenness centrality is considered important as it acts as a bridge or bottleneck for information flow.4. Eigenvector Centrality:Eigenvector centrality measures the influence of a vertex based on the centrality of its neighboring vertices. It assigns a score to each vertex, taking into account the centrality of the vertices it is directly connected to. Eigenvector centrality is calculated iteratively using eigenvalues and eigenvectors. A vertex with a high eigenvector centrality is considered important as it is connected to other highlycentral vertices.Applications of Vertex Centrality:Vertex centrality has numerous applications in various fields, including social network analysis, transportation planning, disease spread modeling, and recommendation systems. By analyzing vertex centrality attributes, researchers and practitioners can gain insights into the structure and dynamics of complex networks.In social network analysis, vertex centrality helps identify key individuals within social networks. For example, degree centrality can be used to identify highly connected individuals who act as hubs of information dissemination. Betweenness centrality can identify individuals who control the flow of information between different communities or groups within a network.In transportation planning, vertex centrality can help identify critical road intersections or public transportation hubs that are crucial for efficient movement of people and goods. Closeness centrality can help determine which locations have better accessibility to other parts of a transportation network.In disease spread modeling, vertex centrality can be used to identify individuals who have a high potential for spreading a disease within a population. By targeting individuals with high centrality, intervention strategies can be optimized to effectively control the spread of the disease.In recommendation systems, vertex centrality can be used to identify influential users or items in a network. By leveraging the centrality of these vertices, personalized recommendations can be generated for users based on their connections and interactions within the network.Conclusion:Vertex centrality attributes play a crucial role in analyzing the importance and influence of vertices within networks. Degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality each provide different perspectives on vertex centrality. The applications of vertex centrality are diverse and span across various domains. By leveraging vertex centrality measures,researchers and practitioners can gain valuable insights into the structure and dynamics of complex networks.。

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