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Extreme HOT regions are CpG-dense promoters in C. elegans and humans

10.1101/gr.161992.113Access the most recent version at doi: published online March 20, 2014Genome Res.Ron A.-J. Chen, Przemyslaw Stempor, Thomas A. Down, et al.humansand C. elegans Extreme HOT regions are CpG-dense promoters in MaterialSupplemental/content/suppl/2014/04/24/gr.161992.113.DC1.htmlP<PPublished online March 20, 2014 in advance of the print journal.Open AccessOpen Access option.Genome Research Freely available online through theLicenseCommons Creative./licenses/by-nc/4.0/License (Attribution-NonCommercial 4.0 International), as described at , is available under a Creative Commons Genome Research This article, published in ServiceEmail Alertingclick here.top right corner of the article or Receive free email alerts when new articles cite this article - sign up in the box at the object identifier (DOIs) and date of initial publication.by PubMed from initial publication. Citations to Advance online articles must include the digital publication). Advance online articles are citable and establish publication priority; they are indexed appeared in the paper journal (edited, typeset versions may be posted when available prior to final Advance online articles have been peer reviewed and accepted for publication but have not yet/subscriptions go to:Genome Research To subscribe to © 2014 Chen et al.; Published by Cold Spring Harbor Laboratory PressResearchExtreme HOT regions are CpG-dense promotersin C.elegans and humansRon A.-J.Chen,Przemyslaw Stempor,Thomas A.Down,Eva Zeiser,Sky K.Feuer, and Julie Ahringer1The Gurdon Institute and Department of Genetics,University of Cambridge,Cambridge CB30DH,United Kingdom Most vertebrate promoters lie in unmethylated CpG-dense islands,whereas methylation of the more sparsely distributed CpGs in the remainder of the genome is thought to contribute to transcriptional repression.Nonmethylated CG di-nucleotides are recognized by CXXC finger protein1(CXXC1,also known as CFP1),which recruits SETD1A(also known as Set1)methyltransferase for trimethylation of histone H3lysine4,an active promoter mark.Genomic regions enriched for CpGs are thought to be either absent or irrelevant in invertebrates that lack DNA methylation,such as C.elegans;however,a CXXC1ortholog(CFP-1)is present.Here we demonstrate that C.elegans CFP-1targets promoters with high CpG density,and these promoters are marked by high levels of H3K4me3.Furthermore,as for mammalian promoters,high CpG content is associated with nucleosome depletion irrespective of transcriptional activity.We further show that highly occupied target(HOT)regions identified by the binding of a large number of transcription factors are CpG-rich pro-moters in C.elegans and human genomes,suggesting that the unusually high factor association at HOT regions may bea consequence of CpG-linked chromatin accessibility.Our results indicate that nonmethylated CpG-dense sequence isa conserved genomic signal that promotes an open chromatin state,targeting by a CXXC1ortholog,and H3K4me3modification in both C.elegans and human genomes.[Supplemental material is available for this article.]Transcription is regulated through functional elements in the ge-nome such as promoters and enhancers.Identifying these geno-mic elements,many of which are recognized by transcription factors(TFs),is a necessary first step in their functional analysis. Toward this goal,the modENCODE and ENCODE Projects have used chromatin immunoprecipitation(ChIP)to map the binding sites for a large number of TFs in C.elegans,Drosophila,and humans (The ENCODE Project Consortium2007,2012;Gerstein et al. 2010,2012;The modENCODE Project Consortium2010;Ne`gre et al.2011;Niu et al.2011;AP Boyle,CL Araya,C Brdlik,P Cayting, C Cheng,Y Cheng,K Gardner,L Hillier,J Janette,L Jiang,et al.,in prep.).Most of the identified TF-binding regions are‘‘low occu-pancy,’’showing binding by one or a few different TFs.As expected of enhancers,low-occupancy sites are enriched for DNA sequence motifs recognized by the bound factors,suggesting direct DNA binding(Gerstein et al.2010;The modENCODE Project Consor-tium2010).In contrast,these mapping studies also identified an unusual class of sites bound by the majority of mapped TFs,termed HOT(Highly Occupied Target)regions.The large number of factors associated with HOT regions is incompatible with simultaneous occupancy,and the regions usually lack sequence-specific binding motifs of the associated factors,indicating that they are unlikely to be classical enhancers(Gerstein et al.2010;The modENCODE Project Consortium2010).Consistent with this idea,Drosophila HOT regions are depleted for annotated enhancers;however, a significant fraction displays enhancer activity in transgenic analyses(Kvon et al.2012).Drosophila and human HOT regions have features of open chromatin,such as nucleosome depletion and high turnover,and in C.elegans they are usually located near genes with ubiquitous expression(Gerstein et al.2010;The modENCODE Project Consortium2010;Yip et al.2012).These previous reports suggest that HOT regions are active genomic ele-ments but fail to provide a clear picture of their function,possibly because of differences in the definition and characterization of these regions.Other DNA sequence features are also known to play regula-tory roles.For example,a hallmark of many mammalian promoters is enrichment for nonmethylated CpG dinucleotides defined as a‘‘CpG island(CGI)’’(Bird et al.1985;Bird1986;Gardiner-Garden and Frommer1987;Illingworth and Bird2009;Deaton and Bird 2011).In contrast to these promoter regions,CpG dinucleotides are sparsely distributed and cytosine-methylated in most other regions in the genome,and differential DNA methylation is thought to be important for the recognition and function of CGIs (Jones et al.1998;Nan et al.1998;Cameron et al.1999;Klose and Bird2006;Joulie et al.2010).Promoter-enriched CpGs have not been observed in invertebrates and are believed to be absent in organisms lacking DNA methylation.Indeed,it is widely consid-ered that the enrichment of CpGs at promoters is a vertebrate-specific phenomenon that requires DNA methylation(Duncan and Miller1980;Gardiner-Garden and Frommer1987;Cooper and Krawczak1989;Ehrlich et al.1990;Antequera and Bird1999; Caiafa and Zampieri2005;Illingworth and Bird2009;Turner et al. 2010;Deaton and Bird2011).In mammals,nonmethylated CpGs are associated with active promoter chromatin features and bound by CXXC1(CFP1),which is part of a SETD1A complex that cata-lyzes methylation of H3K4(Lee and Skalnik2005;Ansari et al. 2008;Butler et al.2008;Tate et al.2010;Thomson et al.2010;Xu Ó2014Chen et al.This article,published in Genome Research,is available under a Creative Commons License(Attribution-NonCommercial 4.0In-ternational),as described at /licenses/by-nc/4.0/.1Corresponding authorE-mail ja219@Article published online before print.Article,supplemental material,and pub-lication date are at /cgi/doi/10.1101/gr.161992.113.Freely available online through the Genome Research Open Access option.24:000–000Published by Cold Spring Harbor Laboratory Press;ISSN1088-9051/14; Genome Research1et al.2011).Although C.elegans lacks DNA methylation,the C. elegans genome encodes a CXXC1ortholog,CFP-1,required for global H3K4trimethylation(Simonet et al.2007;Li and Kelly 2011),suggesting a possible similarity of function.Here we directly compared HOT regions in C.elegans and humans to investigate their functions and ask if these are con-served.We discovered that HOT regions are active promoters enriched for CpG dinucleotides in both organisms.We further show that high CpG density is a feature of many C.elegans pro-moters,and such promoters are targeted by CFP-1.Our results support a conserved function for promoter-dense CpGs in C. elegans and human genomes.ResultsWe collected modENCODE and ENCODE TF mapping data sets for 90C.elegans factors(across developmental stages)and159human factors(from multiple cell types)and determined regions of factor overlap(Supplemental Fig.S1;Methods;The ENCODE Project Consortium2007,2012;Gerstein et al.2010,2012;The modENCODE Project Consortium2010;Ne`gre et al.2011;Niu et al. 2011;AP Boyle,CL Araya,C Brdlik,P Cayting,C Cheng,Y Cheng, K Gardner,L Hillier,J Janette,L Jiang,et al.,in prep).Within each region of TF overlap,we further defined a minimal core region with the highest factor occupancy.This identified35,062C.elegans re-gions bound by1–87factors and737,151human regions bound by 1–138factors.HOT regions are promotersWe first examined chromatin modifications at TF binding regions. We ranked TF binding regions by factor occupancy(high to low) and plotted H3K4me3and H3K4me1as a heat map.As expected, in both organisms,we find that the chromatin of low-occupancy regions has features typical of enhancers(H3K4me3low/ H3K4me1high)(Fig.1).In contrast,as the number of factors in-creases,the chromatin signature becomes more promoter-like (H3K4me3high/H3K4me1low).The majority of TF binding regions in the top5%of occupancy have a promoter-like signature in both organisms,suggesting that they are located at promoters.Consis-tent with this idea,high-occupancy TF binding regions more often overlap with proximal promoter regions compared to low-occu-pancy regions(Fig.1).To increase the power of finding similarities between HOT regions within and between organisms,we analyzed HOT regions in the top1%of occupancy(more than64factors in C.elegans and more than57factors in humans)since these display the strongest enrichment for promoter-like chromatin features in both organ-isms.For comparison,we define‘‘COLD’’low-occupancy regions as those with single-factor binding(representing33.6%of regions in C.elegans and30.8%in humans).The preceding findings suggest that HOT regions may be ac-tive promoters.In C.elegans,the annotated transcript start is often the start of the mature transcript,not the start of transcription,be-cause the primary59end is removed and degraded following trans-splicing(Bektesh and Hirsh1988;Blumenthal1995,2012;Allen et al.2011).To directly investigate whether core HOT regions are functional promoters,we chose10C.elegans core HOT regions of 241–525bp located at various distances upstream of the nearest annotated transcript start(150bp to4.7kb upstream)and cloned each one directly upstream of a GFP reporter gene(Fig.2A).All10 tested HOT regions drove GFP,with most showing widespread expression(Fig.2B;Supplemental Table S1),indicating that the short tested regions are widely active promoters.We also found that genes proximal to HOT regions are usually widely expressed.In C.elegans,78%of genes associated with HOT re-gions are expressed in all tissues assayed using gene expression pro-filing(Spencer et al.2011).Similarly,most(91%)human genes asso-ciated with HOT regions are expressed in all examined cell types in gene expression analyses from the ENCODE Project(see Methods). Taken together,these findings support the view that that most C.ele-gans and human HOT regions are ubiquitously active core promoters. CpG dinucleotides are enriched in C.elegans and human HOT regionsWe next searched for common sequence characteristics of C.elegans and human HOT regions by analyzing their composition of mono-and dinucleotides.In both organisms,HOT regions have high GC content(Fig.3A).Remarkably,C.elegans and human HOT regions show similar patterns of dinucleotide enrichment and depletion, with CG dinucleotides showing the highest enrichment in both organisms(Fig.3A).To further investigate the distribution of CG dinucleotides at HOT and COLD regions,we plotted CpG density across these regions,which showed prominent peaks around the midpoints of HOT regions(Fig.3B,C,upper panel).In contrast, COLD regions do not display these patterns(Fig.3B,C,upper panel). In addition,we find that HOT regions but not COLD regions show strong nucleosome depletion(Fig.3C,bottom panel).The finding that HOT regions are at core promoters raised the possibility that CpG-dense sequences may be a shared promoter feature in C.elegans and human genomes.As described above,CpG enrichment is a known property of the majority of human promoters (Gardiner-Garden and Frommer1987;Illingworth and Bird2009).In these‘‘CpG islands,’’the cytosines of CpGs are typically non-methylated;whereas CpGs are broadly depleted from most regions of the human genome,and nonpromoter CpGs are usually cytosine methylated.Since DNA methylation differences are thought to un-derlie the recognition and function of CpGs as well as the mutation of cytosine that drives depletion of bulk CpGs,it was unexpected to observe enrichment of CpGs at promoters in the C.elegans genome, where DNA methylation and CpG depletion are both absent.The preceding analyses only examined CpG enrichment at narrowly defined HOT regions.To ask whether CpG dinucleotides might be more widely found at C.elegans promoters,we plotted CpG density and observed/expected CpG content(normalizing for local GC content)across protein-coding promoters and found clear CpG enrichment just upstream of the transcription start site(TSS) in many C.elegans promoters,with a narrower distribution in C. elegans than in human(Supplemental Figs.S2,S3;heat map in Fig. 5D,see below).In both humans and C.elegans,genes with CpG-rich promoters more frequently show ubiquitous expression compared to those with CpG-poor promoters(in human68.6%of high CpG and2.3%of low CpG;in C.elegans56.3%of high CpG and21.8%of low CpG;see Methods)(Schug et al.2005;Ramskold et al.2009).Taken together,these findings suggest that CpG di-nucleotides are a feature of widely active C.elegans promoters,as they are in vertebrates.Promoter-dense CpG regions are associated with nucleosome depletionMammalian CpG-rich promoters show strong nucleosome de-pletion that appears at least partially independent of transcriptional2Genome Research Chen et al.activity,suggesting that CpG density plays a role in promoter ac-cessibility (Fenouil et al.2012;Vavouri and Lehner 2012).We therefore investigated the relationship between CpG density and chromatin accessibility at C.elegans promoters.We first separated promoters of highly expressed genes into those with high and low CpG density and compared their nucle-osome distribution patterns.For this analysis,we used ubiqui-tously active C.elegans promoters to avoid effects due to tissue-specific regulation.We found that among these highly active promoters,those with high CpG density are strongly nucleosome depleted compared to those with low CpG density (Fig.4A).We obtained similar results after normalizing CpG content to local GC content (Supplemental Fig.S4A).Therefore,high CpG density but not high transcriptional activity is associated with promoter acces-sibility.To test whether high GC content rather than high CpG density was related to nucleosome depletion,we separated high GC content promoters into those with high or low CpG content and plotted their nucleosome distributions.Promoters with high CpG content show stronger nucleosome depletion than those with low CpG content,indicating that accessibility is linked with high CpG rather than generally high GC content (Supplemental Fig.S4B,C).To investigate whether the level of transcriptional activity affects the nucleosome distribution pattern at promoters,we sep-arated ubiquitously active promoters with high CpG density into different gene expression bands.Ubiquitously active genes are relatively highly expressed,with most (94%)falling into the top 40%of expression when considering all genes.We separated these genes into those in the top and bottom of this range (top 20%andsecond 20%of expression)and compared their nucleosome distributions.The two gene expression classes both show strong nucleosome depletion (Fig.4B),suggest-ing that among ubiquitously expressed genes,the level of transcriptional activity has little effect on the level of nucleosome depletion at promoters with high CpG density.Finally,to investigate whether high CpG density per se can drive nucleosome depletion,we separated all high CpG density promoters into different expres-sion bands.Genes in the middle and bottom expression bands are enriched for tissue-specific expression,whereas those in the top expression band are enriched for being ubiquitously expressed.Pro-moters of all three expression bands dis-play clear nucleosome depletion.How-ever,we found that the level of depletion is lower at promoters of genes in the low/no and middle expression bands com-pared to those of high expression (Fig.4C).This difference suggests that high CpG density alone is not sufficient for strong nucleosome depletion,and that additional regulation can counteract the effects of CpG-rich sequences in pro-moting chromatin accessibility.C.elegans CFP-1is targeted to CpG-rich promoters marked by H3K4me3In higher vertebrates,nonmethylated CpGs are directly recognized by the CXXC domain of CXXC1(CFP1),which leads totri-Figure 1.HOT regions display promoter features.H3K4me3and H3K4me1signals plotted at the centers of core TF overlap regions ranked by the indicated percentile of TF occupancy in humans and C.elegans .(A )For human TF overlap regions,7419(top 1%),30,945(top 2%–5%),a random selection of 100,000of 241,975(top 6%–30%),and 456,812(bottom 70%)regions are plotted.(B )For C.elegans TF overlap regions,376(top 1%),1429(top 2%–5%),9721(top 6%–30%),and 23,536(bottom 70%)regions are plotted.In each TF overlap band,the number of TFs and the percentage of TF core midpoints 6500bp of a TSS are indicated.Scales show linear (human)or log 2(C.elegans )input normalized signalranges.Figure 2. C.elegans HOT regions are functional promoters.(A )The indicated HOT regions (orange)were cloned directly upstream of a histone T GFP fusion gene and examined for the expression of GFP using transgenic reporter assays.(B )Ten of ten assayed regions drove GFP ex-pression.The expression in representative larvae stage 4worms is shown.Coordinates of cloned regions and further information on expression patterns are shown in Supplemental Table S1.HOT regions are CpG-dense promoters in C.elegansGenome Research3methylation of lysine 4of histone H3through recruitment of the SETD1A histone methyltransferase (Lee and Skalnik 2005;Tate et al.2010;Thomson et al.2010).Despite the lack of DNA methylation in C.elegans (Simpson et al.1986),a CXXC1ortholog (CFP-1)con-taining a conserved CXXC domain exists (highlighted in Supple-mental Fig.S5).In addition,it was previously demonstrated that C.elegans CFP-1is required for global H3K4me3levels (Simonet et al.2007;Li and Kelly 2011).These findings raise the possibilitythatFigure 3.HOT regions are enriched for CpG dinucleotides and depleted for nucleosomes.(A )Frequency of the indicated mono-and dinucleotides in HOT (red)and COLD (blue)regions is shown relative to the genome-wide frequency scaled to one (black horizontal line).(B )Heat maps showing the distribution of CpG density in ranked TF overlap regions in human and C.elegans .The color scheme shows the scale (0to 15)for CpG content in a 200-bp window.(C )The distribution of CpG density (top )and nucleosomes (bottom )was plotted for HOT (red)and COLD (blue)regions in the human and C.elegans genomes.Lines show mean signal,darker filled areas show standard error,and lighter filled areas are 95%confidence intervals.All plots show 2-kb regions centered at the midpoint of coreregions.Figure 4.A promoter harboring a CpG dense region is associated with nucleosome depletion in C.elegans .(A )Plots of mononucleosome and CpG distributions across promoters of ubiquitously expressed genes in the top 20%of expression band,separated by high CpG density (red,top 20%)or low CpG density (dark gray,bottom 20%).(B )Mononucleosome and CpG distributions were analyzed for ubiquitously active promoters within 20%of CpG density (protein coding promoters)and separated into the top 20%(red)or second 20%(blue)of expression.(C )The distribution of mononucleosomes and CpGs across promoters with the top 20%CpG density separated into those with high (top 20%,pink),medium (middle 20%,blue),or low/no expression (bottom 40%,orange).Lines show mean signal,darker filled areas show standard error,and lighter filled areas are 95%confidence intervals.Chen et al.4Genome ResearchC.elegans CFP-1might target promoter regions of high CpG density, as in mammals.To test this hypothesis,we mapped the binding sites of CFP-1 by chromatin immunoprecipitation(ChIP)assays of GFP-tagged CFP-1.We find that CpG di-nucleotides are enriched at CFP-1 binding sites and are densely distributed at CFP-1peaks(Supple-mental Fig.S6).Most C.elegans CFP-1binding sites are at pro-moters and overlap regions marked by H3K4me3(Fig.5A–C), consistent with the requirement for CFP-1in generating H3K4me3.Furthermore,the signal intensities of both CFP-1and H3K4me3at promoters show a striking concordance with CpG density(Fig.5D,E).The CFP-1peaks not near annotated59ends also harbor high CpG density and promoter-like chromatin sig-natures(H3K4me3high/H3K4me1low),suggesting that they might identify unknown promoters(Supplemental Fig.S7A).In contrast, little CFP-1signal is found at COLD regions,which usually have an enhancer-like chromatin signature(Supplemental Fig.S7B).We also find that highly expressed genes with high levels of CFP-1(top 20%)show H3K4me3marking and nucleosome depletion,but highly expressed genes with little CFP1(bottom20%)do not, suggesting that CFP-1binding is important for these patterns(Fig. 5E).Furthermore,as with high CpG density,we found that pro-moters with high CFP-1levels show nucleosome depletion and high H3K4me3marking,which was observed in promoters linked with genes in both the top20and second20%expression bands (Fig.5F).We conclude that targeting of CXXC1orthologs to nonmethylated CpG-dense promoters and the relationship with H3K4me3marking is conserved between C.elegans and humans. DiscussionOur results support the view that promoter CpG dinucleotides may function as an ancient conserved promoter signal.Many C.elegans and human promoters show local CpG enrichment and are tar-geted by a nonmethyl CpG binding protein1ortholog(Lee and Skalnik2005and this work),and extreme HOT regions in both organisms are CpG-rich promoters.Furthermore,C.elegans CpG-dense promoters are strongly nucleosome depleted in vivo,as in mammals(Fenouil et al.2012;Vavouri and Lehner2012).How CpG-rich sequences function in promoter accessibility is unclear.One possibility is that the DNA sequence itself initiates accessibility,as it has been observed that selected human CpG-rich promoter sequences poorly assemble nucleosomes in vitro(Ramirez-Carrozzi et al.2009),and large CGIs show low in vitro occupancy (Fenouil et al.2012).If CpG-dense sequence does intrinsically disfavor nucleosome assembly,this could make the regions more available for binding to factors such as CXXC1(CFP1).However,in contrast to the findings of Ramirez-Carrozzi et al.(2009),global in vitro analyses that mapped intrinsically favored nucleosome po-sitions in genomic DNA showed that human CpG-rich promoters are on average enriched for nucleosome occupancy(Valouev et al. 2011).Similarly,we also find that C.elegans CpG-rich promoters display strong nucleosome occupancy in in vitro assembly data sets(Supplemental Fig.S8;Locke et al.2013).These conflicting in vitro results regarding CpG-rich se-quences could be due to differences in the assay conditions used for nucleosome assembly in vitro.Ramirez-Carrozzi et al.(2009) directly tested nucleosome assembly on300-bp fragments, whereas the global in vitro studies(Valouev et al.2011;Locke et al. 2013)assembled nucleosomes on larger fragmented genomic DNAs before micrococcal nuclease treatment to isolate mono-nucleosomes.Another study(Zhang et al.2011)showed that conditions used for nucleosome reconstitution can strongly affect the results obtained,as more in vivo-like nucleosome positions are observed when a salt dialysis method is applied during nucleosome reassembly.Interestingly,the addition of whole-cell extract and ATP in the absence of transcription nearly reconstituted in vivo patterns (Zhang et al.2011),supporting a role for external factors in nucle-osome positioning and occupancy.Further investigations will be required to elucidate the roles of CpG sequences and non-nucleo-some factors in determining nucleosome patterns at promoters.Irrespective of such future studies,our findings suggest a par-adigm whereby CpG-rich sequences promote chromatin accessi-bility,either intrinsically or through the activities of other factors, which facilitates the formation of an active promoter with an open chromatin state.Our results also provide a plausible explanation for the high occupancy of transcription factors at HOT regions. High CpG density at these regions might create and/or maintain highly accessible regions that are available for interactions with TFs and other factors.The finding that C.elegans and human HOT regions are active promoters may seem at odds with the finding that Drosophila HOT regions can function as enhancers in transgenic assays(Kvon et al. 2012).This variance may be partly explained by differences in how HOT regions were defined.We analyzed extreme HOT regions in the top1%of occupancy,whereas Kvon et al.(2012)studied those in the top5%of occupancy.Nevertheless,we find that Drosophila TF binding regions in the top5%of occupancy are enriched for having promoter-like chromatin signatures and frequently overlap transcription start sites(Supplemental Fig.S9A),similar to C.elegans and human HOT regions.We also note that the HOT regions assayed in the Drosophila transgenic study(Kvon et al.2012)were relatively large(a median length of2.1kb)and therefore may harbor multiple regulatory elements.As activity assays were not conducted in the absence of a basal promoter,it is possible that the tested Drosophila regions might contain active promoters as well as enhancers.Interestingly,in contrast to C.elegans and human,we observed only a very small enrichment for CpG dinucleotides in Drosophila HOT regions,with similar enrichment in COLD regions (Supplemental Fig.S9B),suggesting that CpG dinucleotides may not be relevant at Drosophila HOT regions.After CpG,the GC dinucleotide shows the second highest enrichment in human and C.elegans HOT regions(Fig.3),sug-gesting that GpC might play an uncharacterized role at promoters. CpG shows much higher enrichment in humans than in C.elegans, whereas GpC enrichment is similar in the two organisms.This dif-ference might be a consequence of the global depletion of CpG in the nonpromoter regions of the human genome,which is not seen in C.elegans.A possible role for GpC dinucleotides at promoters remains to be characterized.Computational CpG island predictions,which identify DNA sequences with the highest relative CpG density,do not highly overlap promoters in cold-blooded vertebrates(Long et al.2013)or in C.elegans(Supplemental Table S4;Wu et al.2010).Therefore, CpG density alone is unlikely to be the sole signal for an open chromatin state at promoters.Consistent with this,purification of nonmethylated DNA via CXXC binding identifies promoters across a range of vertebrate genomes(Long et al.2013),suggesting that differential DNA methylation may be important for recogni-tion of these regions.As C.elegans lacks DNA methylation,recog-nition of nonmethylated CpG-rich sequence must involve alter-native mechanisms in this organism.An important goal for the future will be to understand how these regions are identified. Given the conservation of CpG promoter enrichment,it seems HOT regions are CpG-dense promoters in C.elegansGenome Research5。

Rational Rose使用手册

可能用到的Rose模型图及元素业务用例、业务参与者、业务工人参与者、用例、关系参与者、用例、关系顺序图、协作图、状态图类图、交互图、状态机图构件图、关系类图、交互图、状态机图、构件图构件图部署图构件图、部署图
2 Rational Rose工具简介
2.1 Rational Rose 2003特性
SVGA兼容显卡（建议256色以上），分辨率 800×600，鼠标（建议两轮以上）
3 Rational Rose 2003企业版的安装
3.2 安装过程
– 安装向导（引导安装） – 安装主程序 – 配置许可协议
4 Rose应用程序界面
– Rose应用程序界面组成
• 应用程序窗口
– 显示载体
• 浏览器窗口
– Rose 2003分企业版、专业版和Rose Modeler三种版本
– Rose 2003支持的特性
• 表1.2
2 Rational Rose工具简介
2.2 Rose的基本功能
• 面向对象建模 • 用例分析 • 支持UML、COM、OMT和Booch’93 • 语义检查 • 支持可控的迭代开发 • 双向工程 • 支持多用户并行开发 • 可以与数据建模工具集成 • OLE链接、自动化 • 多平台可用性
2 Rational Rose工具简介
2.3 Rose的插件及插件程序管理器
– 插件程序管理器（Add-In Manager）
• 控制插件程序的状态：激活状态或非激活状态
3 Rational Rose 2003企业版的安装
3.1 系统要求
项目处理器内存硬盘空间其他
需求
150MHz以上奔腾兼容机 128MB（推荐128MB） 400MB（建议留出最小200MB的交换空间）

Performance of PPM on Terrestrial FSO Links with

468IEEE COMMUNICATIONS LETTERS,VOL.14,NO.5,MAY2010 Performance of PPM on Terrestrial FSO Links withTurbulence and Pointing ErrorsWilfried Gappmair,Member,IEEE,Steve Hranilovic,Senior Member,IEEE,and Erich Leitgeb,Member,IEEEAbstract—In recent work,the error performance of on-off keying has been investigated for free-space optical(FSO)links impaired by both turbulence and pointing loss.In the current letter,the analytical framework is extended to M-ary pulse-position modulation(PPM),providing exact results for M=2. Since the approach is not directly applicable to PPM signals with M>2,a closed-form approximation of the average symbol error probability is derived for this case,which is shown via simulation to be tight over a wide SNR range of interest.Index Terms—Terrestrial FSO communications,pulse-position modulation,turbulent atmosphere,pointing loss.I.I NTRODUCTIONO VER a range of a few kilometers,terrestrial free-space optical(FSO)systems are a serious alternative to radio-frequency(RF)links[1].This is not only due to the fact that the former are not restricted by regulatory bodies,but also because FSO equipment is small,low-cost,and easy to deploy.Furthermore,FSO transmission is secure and not limited by bandwidth considerations.But in terrestrial FSO links,the random changes of the refractive index,caused by temperature inhomogeneities and pressureﬂuctuations,lead to scintillation impairments at the receiver.The gamma-gamma distribution has been shown to be a good model for this sort of fading over a large range of conditions[2].In case of less turbulent channels,a log-normal approximation might be accurate enough[3],[4].In addition to scintillation,the received signal is also corrupted by pointing errors due to building sway[5].This is particularly true in urban areas,where both transmitter and receiver are placed on high-rise structures.Under the assumption of a negligible receiver aperture with respect to the beamwidth,the error performance due to turbulence and misalignment has been studied in[6].Farid and Hranilovic presented a more realistic scenario with non-negligible aper-ture diameters[7].Theirﬁndings were used in[8]to derive the average symbol error probability(ASEP)for on-off keying with K-distributed fading and pointing loss,and in[9]to develop several optimization models for misalignment fading mitigation.Nevertheless,a key challenge in any M-ary pulse-amplitude modulation(PAM)system is that the channel state must be estimated to have optimal threshold values. Manuscript received February4,2010.The associate editor coordinating the review of this letter and approving it for publication was M.Uysal. W.Gappmair is with the Institute of Communication Networks and Satellite Communications,Graz University of Technology,Austria(e-mail: gappmair@tugraz.at).S.Hranilovic is with the Department of Electrical and Computer Engineering,McMaster University,Hamilton,Ontario,Canada(e-mail: hranilovic@mcmaster.ca).E.Leitgeb is with the Institute of Broadband Communications,Graz University of Technology,Austria(e-mail:erich.leitgeb@tugraz.at).Digital Object Identiﬁer10.1109/LCOMM.2010.05.100202An advantage of pulse-position modulation(PPM)is that the optimal detector does not need any information about thechannel state[10].Of course,for the same information rate, M-PPM schemes require M times the bandwidth of M-PAM, but in contrast to RF systems,there is no intrinsic bandwidthlimitation for FSO channels.In this letter,the analytical framework of[8]is extended tocompute the ASEP for2-PPM signals in closed form.Sincethe approach is not applicable to PPM schemes with M>2, a closed-form solution is derived by suitable approximation of the complementary error function.II.C HANNEL M ODELFor M-ary PPM,the transmitted signal x is partitioned into M slots,i.e.,x=(x0,x1,...,x M−1).Assuming that x is distorted by both additive white Gaussian noise(AWGN)and the fading coefﬁcientℎ,the received signal is given by y= (y0,y1,...,y M−1),wherey k=ℎx k+n k.(1) Data symbols are assigned to the slots in x so that exactly one of them has amplitude MP T,while the others are zero.In this context,P T describes the average transmitted optical power, which is limited by eye-safety regulations,and n k denotes the zero-mean AWGN sample with varianceσ2n. Throughout this letter,it is supposed thatℎ=ℎlℎaℎp, whereℎl is the path loss,ℎa the fading due to atmospheric turbulence,andℎp the misalignment because of building sway.Note thatℎl is deterministic(without loss of generality,ℎl=1is assumed throughout this work),whileℎa andℎp are probabilistic with distributions f a(ℎa)and f p(ℎp). Following the results of[7],the pointing loss is approxi-mated byℎp=A0e−2r2/w2e,with r as the radial displacement at the receiver detector due to a mismatch between trans-mitter and receiver,A0=erf2(v)with v=√π/2R a/w b, where erf(⋅)denotes the error function,R a/w b is the radius of the receiver aperture normalized by the beamwaist,and w e=w b[√πerf(v)/(2v e−v2)]1/2represents the equivalent beamwaist.Considering independent and identical Gaussian distribu-tions with varianceσ2s for both horizontal and vertical sway, the probability density ofℎp is given byf p(ℎp)=η2Aη20ℎη2−1p,0≤ℎp≤A0,(2) whereη=w e/(2σs).1089-7798/10$25.00c⃝2010IEEEGAPPMAIR et al.:PERFORMANCE OF PPM ON TERRESTRIAL FSO LINKS WITH TURBULENCE AND POINTING ERRORS469 As mentioned previously,the turbulence-induced fading iswell modeled by a gamma-gamma distribution[7]f a(ℎa)=2(αβ)α+β2Γ(α)Γ(β)ℎα+β2−1a Kα−β(2√αβℎa),ℎa>0,(3)withΓ(⋅)as the standard gamma function and Kν(⋅)the νth-order modiﬁed Bessel function of the second kind[11]. Depending on the atmospheric conditions,the scintillation parametersαandβare speciﬁed as functions of the Rytov variance and the geometry factorπD2a/(2λw L),where L symbolizes the propagation distance,D a=2R a andλw are the aperture diameter and the wavelength,respectively[2]. Following the analytical work in[8]and[12],the channel distribution with gamma-gamma fading and pointing errors is furnished byfℎ(ℎ)=αβη2A0Γ(α)Γ(β)G3,01,3(αβℎA0η2η2−1,α−1,β−1),(4)where G m,np,q (⋅)describes the Meijer G-function1.In contrastto[8],the relationship includes the turbulence parameterβas well;for K-distributed signals(β=1)the expression simpliﬁes accordingly.III.E RROR P ERFORMANCEThe symbol error probability(SEP),conditioned on the fading coefﬁcientℎ,is for2-PPM signals given byp s(ℎ)=12erfc(ℎγs),(5)whereγs=P T/σn and erfc(⋅)is the complementary error function[13].Expressing the latter in terms of Meijer’s G-function,i.e.,erfc(x)=1√πG2,01,2(x210,1/2),as well asapplying the integration theorem for G-functions[14],the ASEP is after some tedious but straightforward manipulations given by(6)on top of the next page.According to[13,Eq.(4.2.53)],the conditional SEP is for M-ary PPM schemes with M>2obtained asp s(ℎ)=1−1√π∫∞−∞[1−12erfc(t)]M−1e−(t−ℎMγs√2)2d t.(7)No closed form could be found by integrating(7)with respect to fℎ(ℎ).On the other hand,numerical solutions of the double integral are daunting and can easily result in accuracy and convergence problems.We consider instead the approximation erfc(x)≈f1(x)= e−x2applied to(7),which yieldsp s(ℎ)≈1−1√π∫∞−∞[1−12e−t2]M−1e−(t−ℎMγs√2)2d t.(8)The bracket term[1−12e−t2]M−1is expanded in binomialform such that the resulting integral ∫∞−∞e−kx2−(x−a)2d x=1Meijer’s G-functions are available as built-in modules in many popular mathematics software packages.xFig.1.Approximations for the complementary error function.√π/(k+1)e−a2k/(k+1)[15,Eq.(3.323/2)]provides after some trivial algebrap s(ℎ)≈M−1∑k=1(−1)k+12k√k+1(M−1k)p s(k,ℎ),(9)withp s(k,ℎ)=e−kℎ2M2γ2s/[2(k+1)].(10) Applying in the next step the identity e−x=G1,00,1(x∣−0)to (10),the relationship can be averaged overℎby using once again the integration theorem for G-functions,which gives p s(k)in(12).Finally,by weighting p s(k)with the k th factor in(9),the approximate ASEP is established asp s≈M−1∑k=1(−1)k+12k√k+1(M−1k)p s(k).(11)It is important to note that there are a great many ap-proximations available for erfc(x).But most of them are not tractable to produce a closed-form solution of(7).Applying erfc(x)≈f2(x)=16e−x2+12e−4x2/3developed in[16],a closed form can be computed,but the computational effort increases signiﬁcantly since the expansion of the bracket term in(8)yields a double sum and not just a single one as in (9).Furthermore,compared to f1(x),it could be veriﬁed that f2(x)produces slightly degraded ASEP results.The reason for this behavior is exempliﬁed in Fig.1:Since the bounds of integration in(7)range from−∞to+∞,the evolution of the approximation to erfc(x)must be considered for negative arguments as well,i.e.,f1(x)performs better because,for x<0,it is closer to erfc(x)than f2(x).In this respect,more accurate approximations of erfc(x),such as those in[17]or [18],would be more promising;however,their use does not provide a simple closed-form approximation of the ASEP.IV.N UMERICAL R ESULTSFig.2presents the ASEP for2-PPM computed via(6)and the approximation in(11)forσs/R a=0.1and different values of w b/R a.Both severe(α=2,β=1)and less severe (α=4,β=2)fading conditions are simulated.It is clear from the plots that(11)forms a fairly tight upper bound for470IEEE COMMUNICATIONS LETTERS,VOL.14,NO.5,MAY2010p s=∫∞p s(ℎ)fℎ(ℎ)dℎ=2α+βη216π3/2Γ(α)Γ(β)G2,56,3(16A20γ2s(αβ)22−η22,1−α2,2−α2,1−β2,2−β2,10,12,−η22)(6)p s(k)=∫∞p s(k,ℎ)fℎ(ℎ)dℎ=2α+βη28πΓ(α)Γ(β)G1,55,2(8kA20M2γ2s(k+1)(αβ)22−η22,1−α2,2−α2,1−β2,2−β20,−η22)(12)1.106sFig.2.Evolution of the ASEP for2-PPM signals(σs/R a=0.1)using Monte Carlo runs(discrete markers),exact solution(6)(dashed lines),and approximation(11)(solid lines).1.106sFig.3.Evolution of the ASEP for16-PPM signals(σs/R a=0.1)using Monte Carlo runs(discrete markers)and approximation(11)(solid lines).ASEPs<0.1.Note also that,for smaller values ofγs,the approximation is no longer an upper bound;this is explained by the fact that e−x2≥erfc(x)for x≥0,while turning into a rather loose lower bound for x<0(see also Fig.1). Nevertheless,for all practical cases of interest simulated in this work,the approximation is able to reliably represent the evolution of error performance.For M=16,the tightness of(11)is also conﬁrmed in Fig.3.The remaining parameters are identical to Fig.2. Since a closed-form solution of the ASEP is not available, Monte Carlo simulations were used to verify the quality of the approximation.Although not shown due to lack of space, similar results hold for other PPM orders as well.V.C ONCLUSIONSThis letter presents a closed-form ASEP for2-PPM FSO channels with gamma-gamma scintillation and pointing errors.For higher order PPM schemes,an accurate estimate has been derived in closed form.Veriﬁed by simulation results,the approximate solution is a tight upper bound for ASEPs<0.1, which is useful to predict the error performance without the need of lengthy simulation runs.R EFERENCES[1]H.Willebrand and B.S.Ghuman,Free-Space Optics:Enabling OpticalConnectivity in Today’s Networks.Indianapolis,IN:Sams Publishing, 2002.[2]M.Uysal,J.Li,and M.Yu,“Error rate performance analysis of codedfree-space optical links over gamma-gamma atmospheric turbulence channels,”IEEE Trans.Wireless Commun.,vol.5,pp.1229–1233,June 2006.[3]K.Kiasaleh,“Performance of coherent DPSK free-space optical com-munication systems in K-distributed turbulence,”IEEE mun., vol.54,pp.604–607,Apr.2006.[4]T.Kamalakis,T.Sphicopoulos,S.S.Muhammad,and E.Leitgeb,“Estimation of the power scintillation probability density function in free-space optical links by use of multicanonical Monte Carlo sampling,”Optics Lett.,vol.31,pp.3077–3079,Nov.2006.[5]X.Liu,“Free-space optics optimization models for building swayand atmospheric interference using variable wavelength,”IEEE Trans.Commun.,vol.57,pp.492–498,Feb.2009.[6]S.Arnon,“Optimization of urban optical wireless communicationssystems,”IEEE Trans.Wireless Commun.,vol.2,pp.626–629,July 2003.[7] A.A.Farid and S.Hranilovic,“Outage capacity optimization for free-space optical links with pointing errors,”IEEE J.Lightwave Technol., vol.25,pp.1702–1710,July2007.[8]H.G.Sandalidis,T.A.Tsiftsis,G.K.Karagiannidis,and M.Uysal,“BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,”IEEE Commun.Lett.,vol.12,pp.44–46,Jan.2008.[9]H.G.Sandalidis,“Optimization models for misalignment fading mit-igation in optical wireless links,”IEEE Commun.Lett.,vol.12,pp.395–397,May2008.[10]K.Kiasaleh,“Performance of ADP-based,PPM free-space optical com-munication systems in atmospheric turbulence,”IEEE mun., vol.53,pp.1455–1461,Sep.2005.[11]M.Abramowitz and I.A.Stegun,Handbook of Mathematical Functions.New York:Dover Publications,1970.[12]H.G.Sandalidis,T.A.Tsiftsis,G.K.Karagiannidis,and M.Uysal,“Optical wireless communications with heterodyne detection over tur-bulence channels with pointing errors,”IEEE J.Lightwave Technol., vol.27,pp.4440–4445,Oct.2009.[13]J.G.Proakis,Digital Communications.New York:McGraw-Hill,1989.[14]V.S.Adamchik and O.I.Marichev,“The algorithm for calculatingintegrals of hypergeometric type functions and its realization in R EDUCE system,”in Proc.Int.Symp.Symbolic and Algebraic Computation, Tokyo,Japan,Aug.1990,pp.212–224.[15]I.S.Gradshteyn and I.M.Ryzhik,Table of Integrals,Series,andProducts.New York:Academic Press,1994.[16]M.Chiani,D.Dardari,and M.K.Simon,“New exponential boundsand approximations for the computation of error probability in fading channels,”IEEE Trans.Wireless Commun.,vol.2,pp.840–845,July 2003.[17]G.K.Karagiannidis and A.S.Lioumpas,“An improved approximationfor the Gaussian Q-function,”IEEE Commun.Lett.,vol.11,pp.644–646,Aug.2007.[18]Y.Isukapalli and B.D.Rao,“An analytically tractable approximationfor the Gaussian Q-function,”IEEE Commun.Lett.,vol.12,pp.669–671,Sep.2008.。

Derek Storey, Business Development Director, ROSEN Headquarters Holger Hennerkes, Corporate

Pipeline Technology Conference 2007Achievement of Long-Term Reliability for Pipeline Systems Authors Rick Meyer, Manager Plant & Facility/ Asset Integrity Management Support, ROSEN Europe B.V. Derek Storey, Business Development Director, ROSEN Headquarters Holger Hennerkes, Corporate Key Account Manager, ROSEN Headquarters Presenter Holger Hennerkes, Corporate Key Account Manager, ROSEN Headquarters, Switzerland 1 AbstractGood industry standards are nowadays available to support a logical and consistent approach to many of the key processes of pipeline integrity management (PIM). Available standards range from supervisory processes such as API 1160 and ASME B31S, to specialized codes for defect assessment such as ASME B31G and RSTRENG. However, every pipeline and every pipeline operator is unique. This means consideration must be given as to how standardized processes can be reconciled with the need for flexibility when dealing with the distinctive features of both pipelines and their owners. Distinguishing features must be acknowledged and supported in the implementation and operation of any PIM system. Recognizing distinctive features and engineering a PIM system to suit individual requirements, means a PIM system needs to be customized to particular pipelines and individual pipeline owners. This paper examines how a PIMS system can be customized to the individual needs of a pipeline operator. It reviews the underlying principles of customization and examines how codified processes, rules and standards can be used during customization to ensure conformity with industry best practices. Moreover, the paper examines the nature, roles and interactions of the three main components of integrity management (IM), people, processes and technology, with the aim of improving the delivered value of the IM program. Finally the associated presentation shows how all these aspects were taken into account in the ROSEN Asset Integrity Management Support (ROAIMS) software. 2 IntroductionPipelines have been the backbone of Europe’s energy transport infrastructure for many decades now and this situation looks certain to continue into the future. Energy consumption within Europe is set to continue its strong upward trend, as shown by estimates from the European Commission (Figure 1). This graph indicates European energy consumption will increase by about 25% during the 30 years from 2000-2030. In the same period energy imports will increase by about 60%, offsetting a decline in European energy production of about 30%. Much of the imported energy will be in the©ROSEN Swiss AG, March 20071Pipeline Technology Conference 2007form of oil and gas, carried mainly by pipelines. It is therefore clear that the future energy security of Europe is highly dependant on the long-term integrity of pipeline systems. The same may be said for other regions of the world. Figure 1: Energy trends in the European Union.Ensuring the long-term integrity of pipelines requires a systematic approach to pipeline maintenance that embodies proven management processes and the judicious application of industry best practices. Industry standards represent the starting point for a range of best practices that are available to all pipeline operators. 3 Best PracticesOne definition of best practices was stated earlier: “best management practices refer to the processes, practices and systems identified in public and private organizations that performed exceptionally well and are widely recognized as improving an organization’s performance and efficiency in specific areas.” [1] From this definition it should be noted that the identification of best processes, practices and systems should not be limited to sources that are close to the company that is carrying out the study. Best practices are ideally sourced on as wide a scale as possible, including a global scale when this is appropriate. Consequently, the research, evaluation and adoption of best practices is strongly tied to concepts of ‘benchmarking’. Benchmarking is nowadays recognized as an essential tool for the continuous improvement of quality[2]. The Xerox Corporation is usually credited with the first major benchmarking project in 1979, when Japanese manufacturers were selling photocopiers in the United States below Xerox's production cost. Xerox was interested to know how Japanese manufacturers produced high quality but less costly photocopier machines. Xerox learned how to increase design and production efficiency and reduce manufacturing costs of its machines from “benchmarking” its Japanese rivals.©ROSEN Swiss AG, March 20072Pipeline Technology Conference 2007Benchmarking goes beyond just analyzing the competition’s products and services. It includes analyzing organizational processes and methods to assess how competitors have achieved their positions. Early benchmarking developments focused on processes and activities, however, the scope has more recently been expanded to include strategies and systems[3]. For typical pipeline operators, PIM best practices tend to fall into five main groups: 1. 2. 3. 4. 5. Statutory rules and regulations (national and local) Compulsory company rules and practices Rules and practices of industry institutions, societies and associations Rules and practices of industry contemporaries, including competitors Rules and practices of industry service companies and product vendorsNational and local statutory rules and regulations head the list of best practice groups because they generally contribute positively to integrity management and they are legally mandatory. The last point is critically important. Any failure to adhere to statutory directives could result in a very serious situation for an operator, especially in the event of a pipeline failure. The combination of a pipeline failure with a failure to adhere to statutes, would tend to significantly increase any penalties sought from an operator and could place the operator under increased scrutiny by a regulator for many years to come. By definition, statutory regulations need to be counted as “best practice” components even when they are not judged to be especially valuable to a particular IM program. Arguably the most important of all best practices in integrity management are the supervisory processes such as API 1160 (liquid pipelines) and ASME B31S (gas pipelines). These two processes are summarized in Figures 2a and 2b.Figure 2a: API 1160 IM process for liquid pipelinesFigure 2b: ASME B31S-2001 IM process for gas pipelines©ROSEN Swiss AG, March 20073Pipeline Technology Conference 20074Integrity Management (IM) FrameworkHigh-level IM frameworks such as API 1160 and ASME B31S (Figures 2a and 2b), may appear to be different at first sight but closer examination reveals many close similarities. To all intents and purposes, the figures can be re-drawn to form a generic model, or typical integrity management process, that can be used for all types of pipelines. A typical, generic form of the IM supervisory process can be drawn as shown in Figure 3. Figure 3: Typical integrity management process and PIMS components.Figure 3 shows that a typical IM supervisory process contains 4 primary components that can be summarized as shown in Table 1: Table 1: Four primary components of a typical integrity management process. 1. Data Management 2. Risk Assessment 3. Reference Inspection, Maintenance and Repair (IMR) Plan 4. Integrity Status and Verification. Management, Analysis & Interpretation of relevant Pipeline Data Determine pipeline threats & probability and consequences of failure Future IMR work planning, scheduling & Budgeting In-field inspection activities and data analysis. Significance of anomalies and mitigation options. Update the IMR Plan©ROSEN Swiss AG, March 20074Pipeline Technology Conference 2007The systematic application of these four main components of integrity management represents one of the most important best practices to be found in the pipeline industry. The principal features of the four components can be described as follows. 5 Data ManagementThe heart of any pipeline integrity management system is the information database. This can be called by a number of names (System Database, Pipeline Network Database, Information Management System, etc.) but essentially it is a repository for all pipeline-related data. The content of the database will be dictated by the operator needs and include the elements described below: • • • • • • • Design and basic pipeline data Manufacturing and construction data Pipeline geometry and route data Relevant operational data Pipeline components Internal inspection and survey data Business/financial information, if required.The design, construction, operation and survey data for a pipeline will, in most instances, have been assembled over many years in a variety of formats and scales. In order to maximise the value of this data to the pipeline operator the key information must be compiled in a logical and accessible data base structure. It must be in a common format, with all drawings, maps and surveys reproduced to a common scale with authenticated datum points to establish the best match between graphical data sets. A best-practice data management process will include an assessment of the available data, reformatting it for input into the system (if required), provision of the appropriate hardware and software and data input: • • • Collection, collation, integration and reformatting of data Provision of a customised information and database (e.g. ROSEN’s Standardized Pipeline Data Warehouse, SPDW) Population of the databaseExpert analysis, based on many years experience, is used to interrogate the compiled data to provide guidance on system integrity and maintenance decisions. Interrogation includes a broad spectrum of functions including component reliability; failure statistics; correlation of pipe, coating and landscape parameters with the incidence of stress corrosion cracking; correlation of third party damage to encroachment, etc.©ROSEN Swiss AG, March 20075Pipeline Technology Conference 20076Risk AssessmentRisk is generally described as the product of the likelihood of a given failure multiplied by the consequence of that event: Risk = Likelihood or Probability of Failure X Consequence of Failure Risk Based Inspection (RBI) is a further example of an industry best practice. RBI is a systematic approach which aims to reduce the overall risk exposure by focussing on the areas of higher risk. This approach reduces the total scope of work and inspection costs in a structured and justifiable way. The role of RBI in maintenance planning is illustrated in Figure 4. Figure 4: The role of RBI in inspection and maintenance planning.Risk based strategies can be applied to pipelines at all stages of their life, from design through to decommissioning. The application of RBI methodologies enables the Operator to: • • • • Identify the primary threats to pipeline integrity, Rank pipelines in terms of risk (probability of failure and consequences), Optimise Inspection, Maintenance, Repair (IMR) activities, i.e. defining the appropriate maintenance need and maintenance activities, and Define an appropriate frequency for conducting the maintenance activity.©ROSEN Swiss AG, March 20076Pipeline Technology Conference 2007A risk-based inspection process can identify potential threats to a pipeline so that further examinations to confirm or reject the findings can be made and corrective actions can be taken. Examples of potential pipeline threats are shown in Figure 5. Figure 5: Potential threats to pipeline integrity.Based on an understanding of pipeline degradation mechanisms and key influencing factors, Rosen has developed a risk assessment methodology that provides a comprehensive semi-quantitative assessment of the risk to pipelines[4]. The primary steps in this assessment include: • • • • • • Data collection and storage in a central database (this includes qualification of data) Segmentation of pipeline into sections (e.g. High Consequence Areas). Consideration of threats, consequences and mitigation to pipeline sections Relative risk assessments “What if” capability for sensitivity analysis Generate report for the reference IMR plan.Risk assessments are often described as being Qualitative or Quantitative. Qualitative assessment implies that absolute numbers are not used to define risk whereas quantitative (or probabilistic) assessment implies that the actual numerical risk level is determined. Generally, in practice a combination of the two approaches that is semiquantitative risk assessment, is adopted by operators to generate a system-wide Inspection, Maintenance and Repair (IMR) plan. This is then supported (Figure 6) by pipeline-specific action plans[5]. The fundamental ways in which a pipeline may fail are generally well understood, for example due to corrosion, fatigue, over-pressure etc. The modes of failure may be controlled by the interaction of several influencing factors. For example susceptibility to failure due to internal corrosion will be influenced by factors such as corrosivity of the transported product, pipeline material resistance, corrosion control mechanisms, condition assessment of the pipeline, etc.©ROSEN Swiss AG, March 20077Pipeline Technology Conference 2007Figure 6: Semi-quantitative risk-based assessment plan.Defining the primary influencing factors and understanding the interaction between these factors is the key to understanding the overall risks. On this basis critical pipelines or “hot-spots” on a particular pipeline can be systematically identified. This information can be used to optimize and plan inspection and maintenance activities and identify the need for further detailed quantitative risk assessment or fitness-for-purpose assessment. For example a programme which is aimed at limiting the risk of external corrosion may involve a combination of external coating and CP surveys, and internal inspection using intelligent pigs to detect and monitor corrosion. 7 Reference Inspection, Maintenance and Repair (IMR) PlanWith confidence in the accuracy and reliability of the data generated by the first two components of the integrity management process, an operator can go forward and make decisions relating to the current and future integrity of a pipeline, remaining life assessment and appropriate preventative maintenance and inspection activities to maintain the design plan for the pipeline[10]. Over many years there has been a strong growth in use of more advanced defect assessment or FFP methods. For example, codes such as RSTRENG, DNV, etc. have gained more widespread application resulting in optimized inspection, maintenance and repair (IMR) strategies. Such methods are fully codified and are an integral part of pipeline integrity guidance documents such as API 1160, ASME 31.8S, and general industry best practice[11]. In combination with risk based inspection methods (RBI), operators can implement optimized and more cost effective IMR approaches while improving pipeline safety and reliability. Figure 7 shows the results of typical FFP analysis of pipeline inspection data.©ROSEN Swiss AG, March 20078Pipeline Technology Conference 2007Figure 7: FFP assessment and investigation / repair planning.It is well known that defect assessment criteria such as ANSI/ASME B31-G can be very conservative and as such may result in unnecessary pipeline repairs. By utilising such alternative integrity assessment approaches (RSTRENG, DNV, etc.) operators significantly benefit from reducing the number of repairs required during future operation. However risk assessment and FFP assessment are only part of the solution, for example, one of the primary parameters which significantly affect any risk assessment, FFP assessment and remnant life study is the determination and application of appropriate corrosion growth rates. The determination of corrosion growth rates enables the determination of the remaining safe working life, timescales for future repairs and re-inspection intervals. By aligning various data sources, pipeline risk assessment, segmentation of pipeline based on physical parameters (wall thickness, location, pressure, etc.) combined with available internal and external inspection data, areas or high integrity risk or “hot-spots” can be identified and the root cause of the corrosion diagnosed (Figure 8). On this basis effort can be focussed on these areas to determine meaningful corrosion growth rate predictions or estimates. Corrosion growth rates can be estimated or predicted using a number of methods depending on the available pipeline data.©ROSEN Swiss AG, March 20079Pipeline Technology Conference 2007Figure 8: Example of pipeline data alignment for corrosion risk analysis and root cause diagnosis.Where no previous in-line inspection data is available, predictive models such as, NORSOK, de Waard and Milliams, etc. combined with engineering judgement can be applied. Where previous in-line inspection data is available, then comparison between repeat inspection runs can be conducted to determine actual corrosion growth rates and compare with such predictive models and the original design plan (Figures 9a and 9b). Figures 9a, 9b: Determination of corrosion growth rates by detailed comparison of repeat inspection runs.Only when the active threats/degradation mechanisms have been identified and the corrosion growth rates have been estimated, can appropriate preventative measures be determined which will form the basis of an appropriate, cost effective corrosion management plan.©ROSEN Swiss AG, March 200710Combined with a review of corrosion management activities, e.g. review of external corrosion protection, CP and coating systems, internal corrosion control/monitoring and correlation with the inspection findings, the primary aim of any integrity management strategy is to diagnose the likely causes of corrosion, and identify appropriate preventative measures to minimise further deterioration of the pipeline. Based on determined corrosion growth rates, predictions of future repairs together with mitigation and re-inspection requirements can be determined.8 Integrity Status and VerificationRosen has developed an integrated approach to support these activities and it is summarized in Figure 10.Figure 10: Integrated approach to FFP Assessment, Corrosion Growth and Integrity Management Strategy Development.Based on available pipeline data, the results that can be achieved from this integrated assessment approach are to:•Provide a qualified statement on the current condition and integrity of the pipeline.•Identify active threats/degradation mechanisms and pipeline risks. Assess probable causes of corrosion.•Recommend appropriate corrosion mitigation and control strategies.•Remaining safe working life.•Timescale for future repairs (pipeline and/or coating).•Re-inspection interval.9 ConclusionsThe preceding sections summarize some of the most important, currently-accepted industry best practices and define an integrity management framework as embodied in various pipeline industry guidance documents for the implementation of a pipeline integrity management system.Based on such a framework, ROSEN’s current integrated approach to pipeline integrity management has been described. In particular, aspects of risk based inspection and maintenance planning and optimization, corrosion growth analysis and integrity assessment, mitigation and prevention have been highlighted.The best practice approach to pipeline integrity management can be summarized as follows:•Identify key performance indicators and use a benchmarking process and gap analysis to define areas for improvement.•Use data management to integrate all relevant pipeline data to support effective decision-making to ensure pipelines are operated safely and cost-effectively.•Use risk assessment to identify any potential risks and plan appropriate maintenance strategies and actions.•Conduct appropriate in-field inspections.•Align and analyze all relevant data sources (data from construction, operations, ILI, CP, soils, etc.)•Conduct defect assessment and corrosion growth analysis to define future damage mitigation and prevention strategies, future repairs and inspection needs.•Conduct remaining life assessment and life extension assessment. •Optimize future inspection, maintenance and repair activities.10 References[1] “Best Practices Methodology - A New Approach for Improving GovernmentOperations”; UnitedStates General Accounting Office (GAO), National Security and InternationalAffairs Division; GAO/NSIAD-95-154; May 1995.[2] “A Review of Literature on Benchmarking”; Dattakumar, R. and Jagadeesh, R.;Benchmarking: An International Journal, Volume 10, Number 3; 2003.[3] “The Theory and Practice of Benchmarking: Then and Now”; Yasin, M.;Benchmarking, Volume 9, Issue 3; 2002.[4] “Advances in pipeline integrity management from an inspection company point ofview”; Storey, D. and Weigold, G.; 17th International Pipeline Pigging & Integrity Management Conference; February 9-10, 2005, Wyndham Greenspoint Hotel,Houston.[5] “Risk Based Integrity Management”; Healy, J., Skinner, I., and Dawson, J.;IATMI Congress & Symposium,13 & 14 November 2001, Jakarta.[6] “The Relevancy of High Quality Input Data for a Condition and/or RiskAssessment”; Meyer, R.E.R. and Hekkema, B.J.; ROSEN Europe B.V.; Pipeline Rehabilitation and Maintenance Conference: Recent Trends in Risk Assessment;11-15 September 2006, Istanbul, Turkey.[7] “The Assessment of Corrosion in Pipelines, Guidance in the Pipeline DefectAssessment Manual (PDAM)”; Cosham, A. and Hopkins, P.; Pipeline Pigging and Integrity Management Conference, Scientific Surveys Ltd., Amsterdam, TheNetherlands, May 2004.。

bruechap007

单个企业的需求
价格
S
价格
P
D
D Q 数量
数量
McGraw-Hill/Irwin
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完全竞争的卖者所面对的需求
如果市场供给增加，市场价格下降。既然每个企业都是价格的接受者，它除了在现有
MR = (TR的变化量) ÷ (Q的变化量) 在由市场决定的价格水平上，MR是不变的—每
新增一单位产出所带来的总收益增加量都是相同的。
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边际成本和短期供给
竞争企业的短期供给曲线是指在一系列价格中的每一价格水平下，企业愿意供给的产出量。
它是指MC曲线上高于停产点的的那部分曲线。由于边际报酬递减规律的存在，它向上倾斜。
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ห้องสมุดไป่ตู้
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短期内的利润最大化
停产
如果价格P下降到低于最小的AVC，那么竞争性企业在短期中将会通过停产来最小化亏损。
如果企业从生产中所得到的总收益低于生产的可变成本，其就会停产。
McGraw-Hill/Irwin
Copyright 2007 by The McGraw-Hill Companies, Inc. All rights reserved.

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The complex separation and extensions of Rokhlin congruence for curves on surfaces

a r X i v :a l g -g e o m /9206009v 1 24 J u n 1992The complex separation and extensions of Rokhlin congruence for curves on surfacesGrigory MikhalkinAbstract The subject of this paper is the problem of arrangement of real algebraic curves on real algebraic surfaces.In this paper we extend Rokhlin,Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on surfaces and give some applications of this extension.For some pairs consisting of a surface and a curve on this surface (in particular for M-pairs)we introduce a new structure —the complex separation that is sepa-ration of the complement of curve into two surfaces.In accordance with Rokhlin terminology the complex separation is a complex topological characteristic of real algebraic varieties.The complex separation is similar to complex orientations in-troduced by O.Ya.Viro (to the absolute complex orientation in the case when a curve is empty and to the relative complex orientation otherwise).In some cases we calculate the complex separation of a surface (for example in the case when surface is the double branched covering of another surface along a curve).With the help of these calculations applications of the extension of Rokhlin congruence gives some new restrictions for complex orientations of curves on a hyperboloid.1Introduction 1.1Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves on surfaces If A is a real curve of odd degree on real projective surface B then A divides B into twoparts B +where polynomial determining A is non-negative and B −where this polynomial is non-positive.V.A.Rokhlin [1]proved the congruence for the Euler characteristic of B −under such strong hypotheses that they follow that B is an M-surface,A is an M-curve and B +is contained in a connected component of B .V.M.Kharlamov [2],D.A.Gudkov and A.D.Krakhnov [3]proved a relevant congruence that makes sense sometimes even if A is not M-but (M-1)-curve,but the hypotheses for B and B +were not weakened.1.2Description of the paperThe results of the paper make sense in the case when a pair consisting of a surface and a curve in this surface is of characteristic type (for deﬁnition see section 2).For pairs of characteristic type we introduce a complex separation of the complement of the curve in the surface that is a new complex topological characteristic of pairs consisting of asurface and a curve in this surface.If the curve is empty then the complex separation is a new complex topological characteristic of surfaces relevant to the complex orientation of surfaces introduced by O.Viro[4].The main theorem is formulated with the help of the complex separation,this theorem is a generalization of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves in surfaces.The main theorem gives nontrivial restrictions even for curves of odd degree on some surfaces.The paper contains also some applications of the main theorem.We prove a congruence modulo32for Euler characteristic of real connected surface of characteristic type.We prove some new congruences for curves on a hyperboloid.We give a direct extension of Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curves on projective surfaces, there we avoid both of the hypotheses that B is an M-surface and B+is contained in a connected component of B.We apply the main theorem to the classiﬁcation of curves of low degrees on an ellipsoid.In particular,we get a complete classiﬁcation ofﬂexible curves on an ellipsoid of bidegree(3,3)(the notion ofﬂexible curve is analogous to one introduced by O.Viro[5]for plane case).One can see that the fact that the theorem can be applied to curves of odd degree follows that this theorem can not be proved in the Rokhlin approach using the double covering of the complexiﬁcation of the surface branched along the curve(since there is no such a covering for curves of odd degree).We use the Marin approach[6].All results of this paper apply toﬂexible curves as well as to algebraic.The author is indebted to O.Ya.Viro for his attention to the paper and consultations. 2Notations and the statement of the main theorem Let C B be a smooth oriented4-manifold such that itsﬁrst Z2-Betti number is zero equipped with an involution conj such that the set R B of itsﬁxed points is a surface. Let C A be a smooth surface in C B invariant under conj and such that the intersection of C A and R B is a curve.These notations are inspired by algebraic geometry.It is said that A is of even degree if C A is Z2-homologous to zero in C B and that A is of odd degree otherwise.It is said that curve A is of type I if R A is Z2-homologous to zero in C A and that A is of type II otherwise.It is said that surface B is of type I abs if R B is Z2-homologous to zero in C B If B is a real projective plane then it is said that B is of type I rel if R B is Z2-homologous to a plane section of C B.We shall say that pair (B,a)is of characteristic type if the sum of R B and C A is Poincar´e dual to the second Stiefel-Whitney class of C B.We shall say that surface B is of characteristic type if R B is a characteristic surface in C B.Let b∗denote the total Z2-Betti number.It is said that manifold C X equipped with involution conj is an(M-j)-manifold if b∗(R X)+2j=b∗(C X)where R X is theﬁxed point set of conj.One can easily see that Smith theory follows that j is a nonnegative integer number.Letσ(M)denote the signature of oriented manifold M.;D M:H∗(M;Z2)→H∗(M;Z2)denote Poincar´e duality operator;[N]∈H∗(M;Z2)denote Z2-homology class of submanifold N⊂M.Let e A=[C A◦C A]C B denote normal Euler number of R B in C B.If Bǫis a surface contained in R B and such that∂Bǫ=R A that we shall denote by e Bthe obstruction to extending of line bundle over R A and normal to R A in C A to ǫthe line bundle over Bǫnormal in C B to R B evaluated on the twisted fundamental class[Bǫ,∂Bǫ]and divided by2.One can see that if(B,A)is a nonsingular pair consisting of an algebraic surface and algebraic curve then e Bǫ=−χ(Bǫ).Letβ(q)denote the Brown invariant of Z4-valued quadratic form q.Theorem1If(B,A)is of characteristic type then there is a natural separation of R B−R A into surfaces B1and B2such that∂B1=∂B2=R A deﬁned by the condition that C A/conj∪B j is a characteristic surface in C B/conj(j=1,2).There is a congruence for the Guillou-Marin form q j on H1(C A/conj∪B j;Z2)e Bj ≡e R B+σ(C B)4−β(q j)(mod8)Addendum1Let q j|H1(R A;Z2)=0a)If A is an M-curve then e Bj ≡e R B+σ(C B)4−βj(mod8)b)If A is an(M-1)-curve then e Bj ≡e R B+σ(C B)4−βj±1(mod8)c)If A is an(M-2)-curve and e Bj ≡e R B+σ(C B)4−βj+4(mod8)then A isof type Id)If A is of type I then e Bj ≡e R B+σ(C B)4(mod4)whereβj is the Brown invariant of the restriction q j|H1(B j;Z2)2.1RemarkSome of components of R A can be disorienting loops in C A(it is easy to see that number of such components is even).Ifαis some1-dimensional Z2-cycle in B j that is a boundary of some2-chainβin R B containing an even number of disorienting C A components of R A then q j(α)=0;if such a number is odd then q j(α)=2.It follows that if(B,A) is of characteristic type then the number of disorienting C A components of R A in each component of R B is even.3The proof of Theorem1and Addendum13.1Calculation of the characteristic class of C B/conjIt is not diﬃcult to see the formula for the characteristic classes of double branched covering:ifπ:Y→X is a double covering branched along Z thenw2(Y)=π∗w2(X)+D−1Y[Z]Applying this formula to p:C B→C B/conj we gettr(Dw2(C B/conj))=Dw2(C B)+[R B]where tr:H2(C B/conj;Z2)→H2(C B;Z2)is transfer(i.e.the inverse Hopf homomor-phism to P).It is easy to see that transfer can be decomposed as the composition H2(C B/conj;Z2)k→H2(C B/conj,R B;Z2)h→H2(C B;Z2)where k is an inclusion homomorphism and h◦k=tr.To prove that h is a monomorphism we use the Smith exact sequence(see e.g.[7]):H3(C B/conj,R B;Z2)γ3→H2(R B;Z2)⊕H2(C B/conj,R B;Z2)α2→H2(C B;Z2)In this sequence theﬁrst component ofγ3is equal to the boundary homomorphism ∂of pair(C B/conj,R B);∂is a monomorphism since H3(C B/conj;Z2)=0(since C B and therefore C B/conj are simply connected).It means that no element of type (0,x)∈H2(R B;Z2)⊕H2(C B/conj,R B;Z2),x=0is contained in Imγ3and there-fore the restriction ofα2to H2(C B/conj,R B;Z2)is a monomorphism and thus H is a monomorphism.Now if[C A]=Dw2(C B)+[R B]thenDw2(C B/conj)=[C A/conj]∈H2(C B/conj,R B;Z2)The exactness of homology sequence of pair(C B/conj,R B)follows since H is a monomor-phism that there exists a surface B1⊂R B such that∂B1=R A and W1=B1∪C A/conj is dual to w2(C B/conj).Let B2be equal to Cl(R B−B1)Surface R B is Z2-homologous to zero in C B/conj since C B is a double covering of C B/conj branched along R B.It follows that W2=B2⊕C A/conj is also a surface dual to w2(C B/conj).Note that the separation of R B into B1and B2is unique sincedim(ker(in∗:H2(R B;Z2)→H2(C B/conj)))=1as it follows from exactness of the Smith sequence.Indeed,since H3(C B/conj;Z2)=0 then this dimension is equal to the dimension of H3(C B/conj,R B;Z2);the dimen-sion of H3(C B/conj,R B;Z2)is equal to1sinceγ4:H4(C B/conj,R B;Z2)→0⊕H3(C B/conj,R B;Z2)is an isomorphism.3.2Proof of the congruence forχ(B j)Note that since W j,j∈{1,2}is a characteristic surface in C B/conj and C B is simply connected we can apply the Guillou-Marin congruence to pair(C B/conj,W j)σ(C B/conj)≡W j◦W j+2β(q j)(mod16)where q j:H1(W j;Z2)→Z4is the quadratic form associated to the embedding of W j into C B/conj(see[8]).Similar to the calculations in[6]we get thate AW j◦W j=2Combining all this we getχ(B j)≡e A4+β(q j)(mod8)Now if q|H1(R A;Z2)=0then additivity of the Brown invariant(see[9])follows thatβ(q j)=β(q|H1(C A/conj;Z2))+βjIt is easy to see that if A is an(M-j)-curve then rkH1(C A/conj;Z2)=j.Points a)and b)of the addendum immediately follow from this.To deduce points c)and d)of theaddendum note that q|H1(C A/conj;Z2)is even iﬀC A/conj is an orientable surface iﬀA isof type I(cf.[6],[9]).4Some applications of Theorem14.1The case C A is empty;congruences for surfaces4.1.1If Dw2(C B)=[R B]then there is deﬁned a complex separation of R B into two closed surfaces B1and B2;there is deﬁned a Z4-quadratic form q on H1(R B;Z2)=H1(B1;Z2)⊕H1(B2;Z2)equal to sum of Guillou-Marin forms of B1and B2which are characteristic surfaces in C B/conj andχ(B j)≡χ(R B)−σ(C B)It follows from an easy observation thatχ(B j)≡β(q|H1(B j;Z2))(mod2)4.1.4RemarkAccording to O.Viro[5]in some cases one can deﬁne some more complex topological char-acteristics on R ly,If B is of type I abs then R B possesses two special reciprocal orientations(so-called semi-orientation)and a special spin structure.If Dw2(C B)=[R B] then R B possesses the special P in−-structure corresponding to Guillou-Marin form q C B: H1(R B;Z2)→Z4of surface R B in C B.The complex separation is a new topological characteristic for surfaces and4.1.1may be interpreted as a formula for this characteristic. Quadratic form q is not a new complex topological characteristic.4.1.5Form q is equal to q C B in the case when these forms are deﬁned(i.e.when Dw2(C B)= [R B])To prove this one can note that the index of a generic membrane in C B bounded by curves in R B diﬀers from the index of the image of this membrane in C B/conj by number of intersection points of this membrane and R B.4.1.6RemarkIf B is a complete intersection in the projective space of hypersurfaces of degrees m j,j= 1,...,s then the condition that Dw2(C B)=[R B]is equivalent to the condition that B is of type I abs in the case when s j=1m j≡0(mod2)and to the condition that B is of type I rel in the case when s j=1m j≡1(mod2).4.2Calculations for double coveringsWe see that to apply Theorem1one needs to be able to calculate the complex separation and the corresponding Guillou-Marin form.We calculate them in some cases in this subsection.By the semiorientation of manifold M we mean a pair of reciprocal orientations of M (note that this notion is nontrivial only for non-connected manifolds).It is easy to see that two semiorientations determine a separation of M;this separation is a diﬀerence of two semiorientations,namely,two components of M are of the same class of separation iﬀthe restrictions of the semiorientations on these components are the same.Let C B be the double covering of surface C X branched along curve C D invariant under the complex conjugation conj X in C X.Suppose C D is of type I.Let C D+be one of two components of C D−R D.Then R D possesses a special semiorientation called the complex semiorientation(see[10]).The invariance of C D under conj X follows that conj X can be lifted in two diﬀerent ways into an involution of C B.Let conj B be one of these two lifts.Let X−=p(R B)),X+=R X−int(X−)where p is the covering map.4.2.1Suppose that Dw2(C B)=[R B].Then for every component C of X+the complex sep-aration of∂C induced from complex separation of R B via p(namely,two circles of∂C are of the same class of separation iﬀthe lie in the image under p of the same class of the separation of R B)is equal to the diﬀerence of the semiorientations on∂C induced by the complex semiorientation of R D and the unique(since C is connected)semiorientation of C.In particular C is orientable.Proofequal to F′∪conj X F′.Then p−1(F)gives an element of H2(C B)(the construction of this element was suggested by O.Viro[11]).Since p−1(F)is invariant under conj B it gives an element in H2(C B/conj B;Z2),say f∈H2(C B/conj B;Z2).Let us calculate the self-intersection number of f.It is easy to see that because of symmetry the self-intersection number of f in C B/conj B is equal to the self-intersection number ofγδin C∪C D+. The deﬁnition of complex semiorientation follows that the self-intersection number ofγδin C∪C D+(and therefore the self-intersection number of f)is equal to zero in theﬁrst case and to one in the second case.4.2.2RemarkIn the case when R X is connected4.2.1completely determines the complex separation of R B.4.2.3(O.Viro[11])Ifλis a loop in R B such that p(λ)=∂(G),where G⊂R X thenq C B(λ)≡2χ(G∩X+)(mod4)4.2.4(O.Viro[11])Supposeγis a path in X−connecting points Qαand Qβof componentsαandβof R D respectively.Then q C B(p−1(γ))=0if the intersection numbers ofγwithαandβare of opposite sign(case1ofﬁg.1)and q C B(p−1(γ))=2otherwise(case2ofﬁg.1)4.3New congruences for complex orientations of curves on ahyperboloidIn this subsection we apply the results of4.1and4.2to double branched coverings over simplest surfaces of characteristic type,a plane and a hyperboloid.To state congruences it is convenient to use the language of integral calculus based on Euler characteristic developed by O.Viro[12].Let R A be a curve of type I in the connected surface R X.We equip R A with one of two complex orientations andﬁx X∞–one of the components of R X−R A.If R X−X∞is orientable and R A is R-homologous to zero for some ring R of coeﬃcient coeﬃcient then there is deﬁned function ind R:R X−R A→R equal to zero on X∞and equal to the R-linking number with oriented curve R A in R X−X∞otherwise.It is easy to see that ind R is measurable and deﬁned almost everywhere on R X with respect to Euler characteristic.Evidently,the function ind R2:R X−R A→R⊗R does not depend on the ambiguity in the choice of one of two complex orientation of R A.4.3.1Consider the case when X=P2,R=Z.Let A be a plane nonsingular real curve of type I given by polynomial f A of degree m=2k.Then R A is Z2-homologous to zero.Let X∞be the only nonorientable component of R P2−R A.Without loss of generality suppose that f A|X∞<0.Deﬁne X±to be equal to{y∈R P2|±f A(y)≥0}.Let p:C B→C P2be the double covering of C P2branched along C A(note that such a covering exists and is unique since m is even and C P2is simply connected).Let conj B:C B→C B be the lift of conj:C P2→C P2such that p(R B)=X−(as it is usual we denote F ix(conj B by R B).Lemmae6.6and6.7of[7]immediately follow that Dw2(C B)=R B.Therefore we can apply4.1.1and4.2.1.Proposition4.2.1follows that p−1(cl(ind Z2−1(2+4Z)))is equal to one of two surfaces of the complex separation of R B.Set B1to be equal to p−1(cl(ind Z−1Z2(2+4Z))),note that B is orientable since ind Z|X∞=0.4.3.2Lemmaβ(q|H1(B1;Z2))≡4χ(ind Z−1(3+8Z)∪ind Z−1(−3+8Z))(mod8)Proof4+β(q|H1(B1;Z2))(mod8)Note that p(R B)=ind Z−1(2Z),χ(ind Z−1(2Z))=1−χ(1+2Z)andσ(C B)=2−2k2. We get4χ(ind Z−1(2+4Z))≡1−χ(ind Z−1(1+2Z))−1+k2+8χ(ind Z−1(±3+8Z))(mod16) We can reformulate this in integral calculus languageR P2ind Z2dχ≡k2(mod16)Thus for projective plane we get nothing new but the reduction modulo16of the Rokhlin congruence for complex orientation[10]R P2ind Z2dχ=k24.3.4Consider now the case when X=P1×P1.Let A be a nonsingular real curve of type I in P1×P1of bidegree(d,r),i.e.the bihomogeneous polynomial f A determining A is of bidegree(d,r)where d and r are even numbers.Let X∞be a component of R P2−R A,X±={y∈R P1×R P1|±f A(y)≥0}.Suppose without loss of generality that f A|X∞<0.Let p:C B→C P2be the double covering of C P2branched along C Aand let conj B:C B→C B be the lift of conj:C P1×C P1→C P1×C P1such that p(R B)=X−.Nonsingularity of A follows that all components of R A non-homologous to zero are homologous to each other.Let e1,e2form the standard basis of H1(R P1×R P1) and let s,t be the coordinates in this basis of a non-homologous to zero component of R A equipped with such an orientation that s,t≥0.If all the components of R A are homologous to zero then set s=t=0.Then R A equipped with the complex orientation produces l′(se1+te2)in H1(R P1×R P1).Note that s and t are relatively prime and l′is even since both d and r are even.4.3.5LemmaIf l′≡0(mod4)then Dw2(C B)=[R B]The proof follows from Lemma3.1of[13].4.3.6LemmaIf l′≡0(mod4)then ind Z4is deﬁned and if sd+tr≡0(mod4)thenR P1×R P1ind Z42dχ≡drNote that the condition that sd+tr≡0(mod4)is just equivalent to the ori-entability of R B.Thereforeβ(q|H1(B1;Z2))≡0(mod4).Further arguments are similarto the plane case,we skip them.4.3.7RemarkThe traditional way of proving of formulae of complex orientations for curve on surfaces (see[15])adjusted to the case when the real curve is only Z4-homologous to zero givesonly congruenceR P1×R P1ind Z42dχ≡dr2(mod8) 4.3.8If l′≡4(mod8)and sd+tr≡2(mod4)thenR P1×R P1ind Z42dχ≡drForm q|H1(B1;Z2)is cobordant to the sum of a form on an orientable surface andsome forms on Klein bottles.The condition that l′≡4(mod8)is equivalent to the condition that the number of forms on Klein bottles non-cobordant to zero is odd.Thusβ(q|H1(B1;Z2))≡2(mod4)4.3.9If l′≡0(mod8)and sd+tr≡0(mod4)thenR P1×R P1ind2Z8dχ≡dr2t+r2(mod8)b)If A is an(M-1)-curve thenχ(B+)≡dr2+4(mod8)then A is of type Id)If A is of type I thenχ(B+)≡0(mod4)This theorem follows from Theorem1and gives some new restriction on the topology of the arrangement of real nonsingular algebraic curve of even bidegree on a hyperboloidwith non-contractible branches.Points a)and b)of4.3.10in the case when d2s≡0(mod2),s+t≡1(mod2)were proved by S.Matsuoka[14]in another way(using 2-sheeted branched coverings of hyperboloid).Point d)of4.3.10is a corollary of the modiﬁcation of Rokhlin formula of complex orientations for modulo4case.4.4The case when A is a curve of even degree on projectivesurface BIn this subsection we deduce Rokhlin and Kharlamov-Gudkov-Krakhnov congruences from Theorem1and give a direct generalization of these congruences.Let B be the surface in P q given by the system of equations P j(x0,...,x q)=0,j= 1,...,s−1,let A be the(M-k)-curve given by the system of equations P j(x0,...,x q)= 0,j=1,...,s−1,where P j are homogeneous polynomials with real coeﬃcients,degP j= m j,s=q−1.Suppose(B,A)is a non-singular pair,R A=∅and m s is even.Denote B+={x∈R B|P s(x)≥0},B−={x∈R B|P s(x)≤0},d=rk(in B+∗:H1(B+;Z2)→H1(R B;Z2)),e=rk(in R A∗:H1(R A;Z2)→H1(R B;Z2)) Set c to be equal to the number of non-contractible in R P q components of R B not intersecting R A.Let us reformulate Rokhlin and Kharlamov-Gudkov-Krakhnov congruences for curve on surfaces in the form convenient for generalization and correcting the error in[1].4.4.1(Rokhlin[1],Kharlamov[2],Gudkov-Krakhnov[3])Suppose B is an M-surface,e=0,B+is contained in one component of R B and in the case when m s≡0(mod4)suppose in addition that c=0.a)If d+k=0thenm1...m s−1m2sχ(B+)≡±1(mod8)4Indeed,it is easy to see that the hypothesis of4.4.1without the condition on c is equivalent to the hypotheses of corresponding theorems in[1],[2]and[3],to reformulate them it is enough to apply the Rokhlin congruence for M-surfaces.4.4.2Remark(on an error in[1])Point2.3of[1]contains a miscalculation of characteristic class x of the restriction to B−of the double covering of C B branched along C A.In[1]it is claimed that if m s≡2 (mod4)then x=w1(B−−A).It led to the omission of the condition on c in both Rokhlin and Kharlamov-Gudkov-Krakhnov congruences.4.4.3Correction of the error in[1]If m s≡2(mod4)then x=in∗αwhere in is the inclusion of B−into R P q andαis the only non-zero element of H1(R P q;Z2)Proofa)If d+k=0thenm1...m s−1m2sχ(B+)≡±1(mod8)4c)If d+k=2andm1...m s−1m2sχ(B+)≡(mod4)44.4.5LemmaThe surface V+=C A/conj∪B+is Z2-homologous to zero in C B/conjpProof-?where p:C Y→C B is the double covering over C B branched along C A and conj Y is such a lift of conj:C B→C B that R Y={y∈C Y|conj Y y=y}is equal to p−1(B−). It is easy to see that this diagram can be expanded to a commutative one by adding φ:C Y/conj Y→C B/conj whereφis the double covering map branched along V+.It follows that[V+]=0∈H2(C B/conj).4.4.6RemarkThe construction of4.4.5allows to deﬁne the separation of R B into B+and B−for any (not necessarily algebraic)curve of even degree and invariant with respect to conj in any (not necessarily projective)complex surface C B equipped with almost antiholomorphic involution conj:C B→C B and such that H1(C B;Z2)=0.In this case there is the unique double covering over C B branched along C A and two possible lifts of conj:C B→C B.The images of theﬁx point sets of these lifts form the desired separation.Note that this separation is diﬀerent from the complex separation.Let us give an internal deﬁnition of this separation.Let x,y be points in R B−R A.Connect these points by pathγinside C B−C A.If the linking number of the loopγconjγand C A is equal to zero then x and y are points of the same surface of the complex separation, otherwise x and y are points of two diﬀerent surfaces of the complex separation.This remark allows to extend4.4.4to the case ofﬂexible curves.4.4.7The proof of4.4.4Note that Dw2(C B)+[R B]=0since Dw2(C B)≡[∞] s−1j=1m j where[∞]∈H2(C B;Z2) is the class of hyperplane section of C B and note that[C A]=0since m s is even.We apply4.1.1,let W be the surface of the complex separation not intersecting B+. Then4.4.5follows that W∪B+is the surface of the complex separation of(B,A)since if W is a characteristic surface in C B/conj then so is W+=W∪V+.Let us prove now that q W+|H1(W;Z2)is equal to q W where q W+and q W are Guillou-Marin forms of W+and W in C B/conj.Note that(q W+−q W)(x),x∈H1(W;Z2)is equal to the linking number of x and V+in C B/conj that is equal to the linking number of x and C A in C B.It is not diﬃcult to see that the condition on c follows that the linking number of x and C A in C B is equal to zero.Theorem1follows thatχ(B+)+χ(W)≡e A4+β(q W)+β(q W+|H1(W+;Z2))(mod8)and4.1.1follows thatχ(W)≡χ(R B)−σ(C B)4+β(q W+|H1(V+;Z2))(mod8)The rest of the proof is similar to that of3.2.4.5Curves on an ellipsoidIn the congruences of4.4.4we was able to avoid the appearance of the complex separation with the help of the separation of R B into B+and B−.For curves of odd degree this does not work since there is no such a separation.Although,if R B is connected then the complex separation does not provide the additional information and still can be avoided. Surfaces B1and B2of the complex separation are determined by the condition that B1∪B2=R B and∂B1=∂B2=R A.The simplest case is the case of curves of odd degree on an ellipsoid.It is well-known that the complex quadric is isomorphic to C P1×C P1,an algebraic curve in quadric is determined by a bihomogeneous polynomial of bidegree(d,r).If the curve is real and the quadric is an ellipsoid then d=r,otherwise the curve can not be invariant under the complex conjugation of an ellipsoid since the complex conjugation of an ellipsoid acts on H2(C P1×C P1=Z×Z in the following way:conj∗(a,b)=(−b,−a) as it is easy to see considering the behavior of conj on the generating lines of C P1×C P1. Thus a real curve on an ellipsoid is the intersection of the ellipsoid and a surface of degree d,this can be regarded as a deﬁnition of real curves on an ellipsoid.4.5.1TheoremLet A be a nonsingular real curve of bidegree(d,d)on ellipsoid B.Suppose that d is odd.a)If A is an M-curve thenχ(B1)≡χ(B2)≡d2+12±1(mod8)χ(B2)≡d2+12+4(mod8) then A is of type Id)If A is of type I thenχ(B1)≡χ(B2)≡1(mod4) Proof4+χ(R B)−σ(C B)2andβj=0because of the triviality of H1(R B;Z2).4.5.2Low-degree curves on an ellipsoidConsider the application of4.5.1to the low-degree curves on an ellipsoid.Gudkov and Shustin[16]classiﬁed real schemes of curves of bidegree not greater then(4,4)on an ellipsoid.To prove restrictions for such a classiﬁcation it was enough to apply the Harnack inequality and Bezout ing the Bezout theorem avoided the extension of such a classiﬁcation to theﬂexible curves.For the curves of bidegree(4,4)one can avoid the using of the Bezout theorem using instead the analogue of the strengthened Arnold inequalities for curve on an ellipsoid.for the classiﬁcation of theﬂexible curves of bidegree(3,3)the old restrictions does not give the complete system of restrictions,they do not restrict schemes2⊔1<2>,3⊔1<1> and2⊔1<1>(see[5]for the notations).Theorem4.5.1restricts these schemes and thus completes the classiﬁcation of the real schemes ofﬂexible curves of bidegree(3,3)on an ellipsoid.4.5.3TheoremThe real scheme of a nonsingularﬂexible curve of bidegree(3,3)on an ellipsoid is1< 1<1>>orα⊔1<β>,α>β,α+β≤4.Each of this schemes is the real scheme of some ﬂexible(and even algebraic)curve of bidegree(3,3)on an ellipsoid4.5.4Curves of bidegree(5,5)on an ellipsoidTheorem4.5.1gives an essential restriction for the M-curves of bidegree(5,5)on an ellip-soid,4.5.1leaves unrestricted18possible schemes ofﬂexible M-curves of bidegree(5,5) and15of them can be realized as algebraic M-curves given by birational transforma-tions of the appropriate aﬃne curves of degree6and another one by Viro technique of small perturbation from the product ofﬁve plane sections intersecting in two diﬀerent points.Thus there are only two schemes of M-curves of bidegree(5,5)unrestricted and unconstructed,namely,1⊔1<6>⊔1<8>and1⊔1<5>⊔1<9>.4.5.5The case of bidegree(4,4)The classiﬁcation of the real schemes of curves of bidegree(4,4)[16]shows that there is no congruence like4.5.1for curves of even degree:the Euler characteristic of surfaces B1 and B2for curves of bidegree(4,4)can be any even number between-8and10.Thus the condition that Dw2(C B)+[R B]+[C A]=0is essential.4.5.6Addendum,the Fiedler congruence for curves on an ellipsoidLet A be an M-curve of bidegree(d,d)on ellipsoid B.Suppose that d is even,the Euler characteristic of each component of B1is even andχ(B1)≡2(mod4)thenχ(B2)≡d2(mod16)χ(B1)≡2−d2(mod16)Proof。

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178
Chapter 7
Discrete Probability
CHAPTER 7 Discroduction to Discrete Probability
2. The probability is 1/6 ≈ 0.17, since there are six equally likely outcomes. 4. Since April has 30 days, the answer is 30/366 = 5/61 ≈ 0.082. 6. There are 16 cards that qualify as being an ace or a heart, so the answer is 16/52 = 4/13 ≈ 0.31. We could also compute this from Theorem 2 as 4/52 + 13/52 − 1/52. 8. We saw in Example 11 of Section 6.3 that there are C (52, 5) possible poker hands, and we assume by symmetry that they are all equally likely. In order to solve this problem, we need to compute the number of poker hands that contain the ace of hearts. There is no choice about choosing the ace of hearts. To form the rest of the hand, we need to choose 4 cards from the 51 remaining cards, so there are C (51, 4) hands containing the ace of hearts. Therefore the answer to the question is the ratio C (51, 4) 5 = ≈ 9.6% . C (52, 5) 52 The problem can also be done by subtracting from 1 the answer to Exercise 9, since a hand contains the ace of hearts if and only if it is not the case that it does not contain the ace of hearts. 10. This is similar to Exercise 8. We need to compute the number of poker hands that contain the two of diamonds and the three of spades. There is no choice about choosing these two cards. To form the rest of the hand, we need to choose 3 cards from the 50 remaining cards, so there are C (50, 3) hands containing these two speciﬁc cards. Therefore the answer to the question is the ratio C (50, 3) 5 = ≈ 0.0075 . C (52, 5) 663
Section 7.1
An Introduction to Discrete Probability
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16. Of the C (52, 5) = 2,598,960 hands, 4 · C (13, 5) = 5148 are ﬂushes, since we can specify a ﬂush by choosing a suit and then choosing 5 cards from that suit. Therefore the answer is 5148/2598960 = 33/16660 ≈ 0.0020 . 18. There are clearly only 10 · 4 = 40 straight ﬂushes, since all we get to specify for a straight ﬂush is the starting (lowest) kind in the straight (anything from ace up to ten) and the suit. Therefore the answer is 40/C (52, 5) = 40/2598960 = 1/64974 . 20. There are 4 royal ﬂushes, one in each suit. Therefore the answer is 4/C (52, 5) = 4/2598960 = 1/649740 . 22. There are 100/3 = 33 multiples of 3 among the integers from 1 to 100 (inclusive), so the answer is 33/100 = 0.33 . 24. In each case, if the numbers are chosen from the integers from 1 to n , then there are C (n, 6) possible entries, only one of which is the winning one, so the answer is 1/C (n, 6). a) 1/C (30, 6) = 1/593775 ≈ 1.7 × 10−6 b) 1/C (36, 6) = 1/1947792 ≈ 5.1 × 10−7 c) 1/C (42, 6) = 1/5245786 ≈ 1.9 × 10−7 d) 1/C (48, 6) = 1/12271512 ≈ 8.1 × 10−8 26. In each case, if the numbers are chosen from the integers from 1 to n , then there are C (n, 6) possible entries. If we wish to avoid all the winning numbers, then we must make our choice from the n − 6 nonwinning numbers, and this can be done in C (n − 6, 6) ways. Therefore, since the winning numbers are picked at random, the probability is C (n − 6, 6)/C (n, 6). a) C (34, 6)/C (40, 6) = 1344904/3838380 ≈ 0.35 b) C (42, 6)/C (48, 6) = 5245786/12271512 ≈ 0.43 c) C (50, 6)/C (56, 6) = 15890700/32468436 ≈ 0.49 d) C (58, 6)/C (64, 6) = 40475358/74974368 ≈ 0.54 28. We need to ﬁnd the number of ways for the computer to select its 11 numbers, and we need to ﬁnd the number of ways for it to select its 11 numbers so as to contain the 7 numbers that we chose. For the former, the number is clearly C (80, 11). For the latter, the computer must select four more numbers besides the ones we chose, from the 80 − 7 = 73 other numbers, so there are C (73, 4) ways to do this. Therefore the probability that we win is the ratio C (73, 4)/C (80, 11), which works out to 3/28879240, or about one chance in ten million (1.04 × 10−7 ). The same answer can be obtained by counting things in the other direction: the number of ways for us to choose 7 of the computer’s predestined 11 numbers divided by the number of ways for us to pick 7 numbers. This gives C (11, 7)/C (80, 7), which has the same value as before. 30. In order to specify a winning ticket, we must choose ﬁve of the six numbers to match ( C (6, 5) = 6 ways to do so) and one number from among the remaining 34 numbers not to match ( C (34, 1) = 34 ways to do so). Therefore there are 6 · 34 = 204 winning tickets. Since there are C (40, 6) = 3,838,380 tickets in all, the answer is 204/3838380 = 17/319865 ≈ 5.3 × 10−5 , or about 1 chance in 19,000 . 32. The number of ways for the drawing to turn out is 100 · 99 · 98. The number of ways of ways for the drawing to cause Kumar, Janice, and Pedro each to win a prize is 3 · 2 · 1 (three ways for one of these to be picked to win ﬁrst prize, two ways for one of the others to win second prize, one way for the third to win third prize). Therefore the probability we seek is (3 · 2 · 1)/(100 · 99 · 98) = 1/161700. 34. a) There are 50 · 49 · 48 · 47 equally likely outcomes of the drawings. In only one of these do Bo, Colleen, Jeﬀ, and Rohini win the ﬁrst, second, third, and fourth prizes, respectively. Therefore the probability is 1/(50 · 49 · 48 · 47) = 1/5527200 . b) There are 50 · 50 · 50 · 50 equally likely outcomes of the drawings. In only one of these do Bo, Colleen, Jeﬀ, and Rohini win the ﬁrst, second, third, and fourth prizes, respectively. Therefore the probability is 1/504 = 1/6250000.