A note on fructification of Plagiozamites Zeiller (Noeggerathiales) based on association

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慷慨援助之举的英语作文

In the small town where I live,there is a story of generosity that has become a legend among the residents.It all started a few years ago when a local businessman,Mr. Smith,noticed that many children in the community were struggling with their education due to a lack of resources.Mr.Smith,a man of great wealth and influence,decided to take action.He established a foundation with the sole purpose of providing financial aid to underprivileged children in our town.The foundation offered scholarships,tutoring services,and even school supplies to those in need.The impact of Mr.Smiths generosity was immediate and profound.Children who had previously struggled academically began to excel,gaining confidence and a sense of hope for the future.Parents were overwhelmed with gratitude,as they saw their childrens lives change for the better.One particular story stands out.A young girl named Emily was struggling in school, unable to keep up with her classmates due to a learning disability.Her family could not afford the specialized tutoring she needed.When Mr.Smiths foundation learned of her situation,they stepped in and provided the necessary support.Emilys grades improved dramatically,and she went on to graduate at the top of her class.The ripple effect of Mr.Smiths generosity extended beyond the children he directly helped.Local businesses began to thrive as families gained stability and were able to contribute more to the economy.The towns overall educational performance improved, attracting new families and investment.Mr.Smiths act of kindness did not go unnoticed.He was honored with numerous awards and recognitions for his philanthropy.However,the true measure of his generosity lies in the lives he changed and the community he helped to transform.In a world where selfishness and greed often dominate the headlines,Mr.Smiths story serves as a powerful reminder of the difference one person can make when they choose to give selflessly.His legacy will continue to inspire future generations to follow in his footsteps and make the world a better place,one act of kindness at a time.。

Note on a Theorem of Munkres

A NOTE ON A THEOREM OF MUNKRES
arXiv:math/0403055v2 [math.GT] 14 Mar 2004
EMIL SAUCAN Abstract. We prove that given a C ∞ Riemannian manifold with boundary, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result holds extends to C 1 manifolds, and that in dimensions 2, 3 and 4 it also holds for P L manifolds. We employ the main result to prove that given any orientable C ∞ Riemannian manifold with boundary admits quasimeromorphic mappings onto Rn . In addition some generalizations are given.
1. Introduction The existence of triangulations for C 1 manifolds without boundary is known since the classical work of Whithead ([31], 1940). This result was extended in 1960 by Munkres ([17]) to include C r , 1 ≤ r ≤ ∞, manifolds with boundary. To be more precise, he proved that any C r triangulation of the boundary can be extended to a C r triangulation of the whole manifold. Yet earlier, in 1934-1935 (even before Whithead’s work) Cairns1 ([3], [4]) proved the existence of fat triangulations for compact C 1 manifolds and for manifolds with boundary having a ﬁnite number of compact boundary components. Moreover, his triangulations were fat (see deﬁnition bellow), something which Munkres’ method achieved only away from the boundary (see Section 2.2). Unfortunately, it seems that little interest existed during the following decades, for studying generalizations of the results above ([8] representing a notable exception). The interest in the existence of fat triangulation was rekindled by the study of quasiregular and quasimeromorphic functions, since the existence of fat triangulations is crucial in the proof of existence of guasiregular (quasimeromorphic) mappings (see [15], [29]) and in 1992 Peltonen ([18]) proved the existence of fat triangulations for C ∞ Riemannian manifolds, using methods partially based upon another technique of Cairns (originally developed for triangulating manifolds of class ≥ C 2 ). In this paper we extend Munkres’ Theorem to the case of fat triangulations of manifolds (and orbifolds – see Section 5.2.) with or without boundary and we show how to apply this main result in order to prove the existence of quasimeromorphic functions. Our main result is the following Theorem: Theorem 1.1. Let M n be an n-dimensional C ∞ manifold with boundary. Then any fat triangulation of ∂M n can be extended to a fat triangulation of M n .

小学上册第十四次英语第1单元暑期作业

小学上册英语第1单元暑期作业英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The chef creates new _____ (菜肴).2.The ________ in my room is very soft.3.What is the main purpose of a bridge?A. To connect two sidesB. To support vehiclesC. To provide a viewD. To carry waterA4.I enjoy playing ______ outside.5.What do we call the place where animals live in the wild?A. ZooB. HabitatC. FarmD. CircusB Habitat6.We have a ______ (有趣的) teacher.7.The beaver builds a _______ to live in.8.In chemistry, a _______ is a shorthand way to represent a chemical substance. (化学符号)9.Hawaii is an example of a __________.10.Which month is Christmas celebrated?A. NovemberB. DecemberC. JanuaryD. OctoberB11.My brother is my best _______ who plays with me every day.12.What do we wear on our feet?A. HatB. ShoesC. GlovesD. Scarf13.We found a __________ (化石) in the ground.14.The ____ makes a loud sound and is often found in the barn.15.The chemical formula for zirconium oxide is ______.16.The __________ is a place where animals live in their natural habitat.17.The __________ (历史的主题) resonate across cultures.18.Wildflowers grow __________ (自然) in meadows.19.The ______ loves to explore new places.20. A __________ is formed through the interaction of various geological processes.21.What is the capital of China?A. ShanghaiB. BeijingC. Hong KongD. TaipeiB22.The ancient Greeks held ________ every four years.23.What do you call the time of day when the sun rises?A. MorningB. NoonC. EveningD. NightA24. A _______ is a reaction that occurs in the laboratory.25.The snow is _______ (白色的).26.The ________ (sunset) is beautiful.27.What is the main language spoken in Spain?A. SpanishB. FrenchC. ItalianD. Portuguese28.I want to learn ________ (游泳).29.The dog is _____ at the ball. (looking)30.The chair is ________.31.What do you call the part of the plant that absorbs water and nutrients?A. StemB. LeafC. RootD. FlowerC32.Atoms are made up of protons, neutrons, and _____.33. (Ming) Dynasty is known for its trade and exploration. The ____34.The atomic mass of an element is the average mass of its __________.35.What is the name of the famous bridge in San Francisco?A. Brooklyn BridgeB. Golden Gate BridgeC. London BridgeD. Sydney Harbour BridgeB Golden Gate Bridge36.My _____ (弟弟) loves to play with his toy trains. 我弟弟喜欢玩他的玩具火车。

英语作文墨家思想

Mozi,an ancient Chinese philosopher,founded the Mohist school of thought,which was one of the major philosophical schools during the Warring States period475221BC. His teachings,which are contained in the Mozi text,are characterized by a focus on universal love,utilitarianism,and opposition to aggression.Here is an essay on Mohist philosophy,highlighting its key principles and their relevance to modern society.Title:The Timeless Wisdom of MohismIntroductionIn the vast expanse of Chinese philosophical thought,the Mohist school stands out for its unique emphasis on universal love jian ai,utilitarianism li yi,and opposition to unjust wars.Mozi,the founder of this school,was a contemporary of Confucius and Laozi,and his teachings offer a different perspective on how society should be organized.This essay delves into the core tenets of Mohism and explores their enduring value in contemporary society.Universal Love Jian AiThe central concept of Mohism is universal love,which is the idea that one should love all people equally,without distinction of status or relationship.Mozi argued against the Confucian concept of graded love,which prioritizes family and social hierarchy.He believed that the lack of universal love was the root cause of social strife and conflict.By promoting equality in love,Mohists sought to create a harmonious society where everyones wellbeing is considered.Utilitarianism Li YiMohists were utilitarians,advocating for actions that maximize overall benefit and minimize harm.This principle was applied to all aspects of life,from personal conduct to statecraft.Mozi emphasized the importance of practicality and efficiency,arguing that actions should be judged by their outcomes rather than by adherence to traditional rites or customs.This pragmatic approach to ethics has parallels in modern utilitarian philosophy, which seeks to make decisions based on the greatest good for the greatest number. Opposition to Aggression Fei GongMozi was a staunch pacifist,opposing all forms of aggression,especially unjust wars.He believed that war was a great evil,causing unnecessary suffering and destruction.To promote peace,Mozi advocated for a defensive military strategy and the strengthening ofalliances among states.His ideas on nonaggression have influenced later thinkers and can be seen as an early form of internationalism,emphasizing cooperation and mutual respect among nations.Innovation and TechnologyAnother aspect of Mohist thought was a strong emphasis on innovation and the application of technology for the benefit of society.Mohists were known for their contributions to science and engineering,including advancements in defensive military technology.This focus on practical knowledge and its application to improve human life is a testament to the forwardthinking nature of Mohism.ConclusionThe Mohist school of thought,with its principles of universal love,utilitarianism,and opposition to aggression,offers a unique perspective on how society can be organized for the common good.While rooted in the historical context of ancient China,the teachings of Mozi resonate with modern concerns about social justice,ethical decisionmaking,and international relations.As we continue to grapple with the challenges of our time,the wisdom of Mohism reminds us of the importance of empathy,practicality,and peace in building a better world.ReflectionIn reflecting on Mohism,one cannot help but be struck by its relevance to contemporary issues.The call for universal love and the rejection of violence as a means to resolve conflicts are messages that are as pertinent today as they were in Mozis time.As we navigate the complexities of our interconnected world,the Mohist philosophy serves as a reminder of the potential for a more harmonious and equitable society.。

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Initial Remark of France

Initial Remark of FranceBy French Delegation to UNSC of IMUNC2011Honorable Chair and distinguished delegates,France has been a major participant in many important international affairs, and we have always been devoting our strength to the maintenance of peace and stability, promotion of international cooperation and economic development, anti-terrorism and anti-proliferation of mass destruction weapons, and protection of human rights. Now, we are glad to have the chance to work with you all to address crucial international issues.As a European country and a major promoter of the European integration process, France stresses the necessity of unity and cooperation of European countries when addressing important issues.Now there are some certain issues that concern all the countries in the world significantly, and they are also closely connected to the national interest of France.Firstly, Somalia has been a concern to all of us. The recent development of the situation in Somalia is not so hopeful. The country has been hit by the most serious drought in the last 60 years, which resulted in significant crop failures. Now, famine has swept across the country, and people there are starving. To make the situation even worse, the most seriously affected areas are currently under the control of an insurgent group with direct connection with the notorious terrorist group al-Qaeda, namely al-Shabab, and they have denied the entry of most humanitarian aid.France expresses its deep concern about the situation there. At this moment, we deem immediate humanitarian aid to be necessary to keep thousands of Somalis from starvation. Now combined with the threat of al-Shabab to disrupt the delivery of humanitarian aid, Somali pirates are threatening our supply line by sea. Therefore, escort for humanitarian missions is essential from our point of view. The Operation Atlanta of the EU has been providing escort for World Food Program (WFP) vessels for a long time, and we call upon relevant states to provide necessary assistance. Because of the famine, tens of thousands Somalis have fled their homes to neighboring countries, and they became a huge burden for the limited financial resources of those countries, so we recommend that the UN provide financial and material aid to those countries, and humanitarian aid should be delivered to those refugees as well. Also, we need further information about the situation in Somalia, so an investigation team to Somalia will be really helpful.Piracy is another major problem. The pirates have posed a serious threat on merchant ships, WFP vessels, and all ships passing by alike. Now, many states have sent naval forces to protect vessels sailing past Somali waters. France appreciates the efforts of those states and the protection they provided. We also wish to express our appreciation of the coordination efforts of Contact Group of Piracy off the Coast of Somalia. Taking note of the inability of the Transitional Federal Government (TFG) to prosecute and detain suspects caught in Somali waters, wewelcome the proposal of setting up a specialized court for piracy and armed robbery at sea off the coast of Somalia. We also recommend relevant countries revise their domestic laws to criminalize piracy and armed robbery at sea, in order to bring the guilty to justice. We have also noticed that some pirates are using the illegal fishing and dumping, including toxic substance, in Somali waters as their justification for their acts, therefore we call upon relevant states to protect the coast of Somalia, and we suggest that the Somali authority establish its own coast guard aimed at countering piracy and protecting the coast of Somalia from illegal acts.A direct cause of all these problems mentioned above, in our opinion, is the lack of a functioning central government. France recognizes the TFG and Transitional Parliament as the legitimate representative of Somali people, but we have noticed that Somali territory is largely dominated by numerous self-governing clan leaders. The term of the TFG and Transitional Parliament is going to expire in August, and France recommends that Somalis should form a new coalition government, and include those clan leaders into the political system, as a mean to extend the authority of the central government. Also, international aid and supervision will be necessary for the forming of the new government and parliament.The insurgency in the southern part of the country also weakens the power of the central government, and it now threatens the delivery of humanitarian aid. France strongly condemns the terrorist activity of al-Shabab, and deems it as a terrorist group with direct al-Qaeda connection. The existence of this kind of group should be illegal both domestically and internationally, and they must be ultimately eliminated unless they make proper statement, cease fire and stop terrorist activity, and sever all connection with al-Qaeda. France appreciates the peace-keeping efforts of forces of the UN and AU, and we are willing to do our part to guarantee the safe passage of humanitarian aid, to protect civilians, and to restore order to Somalia.France believes that through the perseverance of Somali people and international community, peace, prosperity and democracy can finally be achieved in Somalia.Secondly, the situation in the Middle East has been another focus of ours, and now the recent political upheavals have made the situation there even more complicated. As a crucial part of the Middle East, and a core of all the conflicts, the Palestinian Question has always remained a puzzle to the international community.France is gravely concerned about the humanitarian situation in Palestine, especially in Gaza Strip, which is now under blockade of Israel from land and sea. Reports have shown that the livelihood of the inhabitants of Gaza is just unacceptable and unsustainable. Noting with concern the past actions of Israel blockading forces and the certain items banned from entry by Israel, we deem it inappropriate to maintain a blockade at this level, and the blockade is a direct cause of the unsatisfactory humanitarian condition in Gaza in our view. Francereiterates its obligation to protect fundamental human rights, and we deem it our duty to urge Israel to ease the blockade, and to allow humanitarian aid to pass unhindered. We also recommend that the UN send an investigation team into Gaza to collect more information about the situation there.However, bearing in mind the safety and interest of Israeli people, we suggest that all the humanitarian aid should be subject to examination by the UN and Israeli personnel. Also, we propose a complete arms embargo on Gaza, aimed at depriving the terrorist groups operating in Gaza of their ability to launch attacks on Israeli targets.As for the Palestinian refugee problem, France stresses that they have the legal rights to return to their homeland and claim compensation, in accordance with the UNGAR194. We affirm that we are willing to provide necessary humanitarian aid to Palestinians living in refugee camps in neighboring countries of Palestine, and call upon relevant countries to take actions to protect the basic rights of Palestinian refugees and to improve their livelihood.France supports the partition plan of founding an independent, viable, sovereign Palestinian state alongside Israel on occupied Palestinian territories, in accordance with UNGAR181. We also recognize the Palestinian Authority as the sole legitimate representative of Palestinian people. Believing the mutual dialogue between Palestinian Authority and Israel will be necessary when facing major emergency, we recommend that Israel and Palestinian Authority form a joint committee composed of senior representatives of both sides to handle major crisis.We reaffirm that France will continue to support the cause of Palestinian people, and will continue providing humanitarian aid to Palestine, as well as equipment and training for Palestinian security forces.On the issue of anti-terrorism, we reiterate that France is totally against any form of terrorism, and regards all Palestinian organizations that deny the right of Israel to exist in peace and are conducting any kinds of attacks on civil or military targets of Israel as terrorist groups. Bearing in mind our obligation to maintain peace and stability in the region, we will do our utmost to help Government of Israel and Palestinian Authority to fight against terrorism, and we call upon both sides to work together to address this issue.One major obstacle to the current peace process is the territorial disputes between Israel and neighboring countries. France recognizes the pre-1967 borders, which is also called Blue Line, and has been recognized by the UN. Keeping in mind the interest of people living in the region, we urge relevant countries and regions to settle the disputes by negotiation, rather than by force and military means. Aware of Israeli’s concern about their safety once Israel evacuates from the occupied lands, we suggest the UN send peace-keeping forces to prevent terrorist attacks on the borders.We strongly recommend Israel and Palestinian Authority resume peace process and start high-level talks immediately. In the hope of achieving a just, durable peace in the Middle East, France is willing to do its part.。

PlagiarismRation...

PlagiarismRationale: Plagiarism demonstrates a lack of integrity and character that is inconsistent with the goals and values of Toms River High School East.Excellent written expression of well-formulated ideas is a fundamental skill for academic and career success. Plagiarism interferes with the assessment and feedback process that is necessary in order to promote academic growth. Plagiarism defrauds the instructor with a false view of a student’s strengths and weaknesses. It may prevent further instruction in areas of weakness and delay the student in reaching his or her potential.Plagiarism includes:-taking someone else’s assignment or portion of an assignment and submitting it as your own-submitting material written by someone else or rephrasing the ideas o f another without giving the author’s name or source-presenting the work of tutors, parents, siblings, or friends as your own-submitting purchased papers as your own-submitting papers from the Internet written by someone else as your own-supporting plagiarism by providing your work to others, whether you believe it will be copied or not(Please NOTE: There is no distinction in terms of the amount or percentage of plagiarism in a paper; if any plagiarism is present, the entire paper will be deemed as tainted.)CheatingRationale: Cheating demonstrates a lack of integrity and character that is inconsistent with the goals and values of Toms River High School East.Education is based on learning specific skills, forming lifelong work habits, and developing mature coping skills according to each student’s unique abilities. Stress sometimes propels students to make unethical choices. When students choose to cheat, it may be a symptom of more serious problems such as inappropriate class placement, over-commitment to extra-curricular activities, and/or academic desperation. The compromise of their values through cheating may lead to loss of self-esteem, as the students are often painfully aware of their shortcomings and fight a tiring battle to preserve their images at the cost of their ethics. True self-esteem is based on competence. Cheating robs students of their opportunity to become competent. Additionally, cheating harms other students in a ripple effect by artificially raising the cheater’s GPA and class rank, possibly qualifying the cheater for undeserved and unearned honors, awards, and scholarships at the expense of honest students. All students are expected to produce their own work except on projects designated by the teacher as cooperative efforts. Teachers will indicate which assignments are to be cooperative efforts and will establish guidelines for the use of such aids as calculators, computers, word processors and published study guides. If a student is in doubt about the ethical standards applicable to a particular situation, the student is responsible for clarifying the matter with the teacher.Cheating includes:-copying, faxing, emailing, or in any way duplicating assignments that are turned in, wholly or in part, as original work-exchanging assignments with other students, either handwritten or computer generated, whether you believe they will be copied or not-using any form of artificial memory aid during tests or quizzes without the expressed permission of the instructor-using a computer or other means to translate an assignment from one language into another language and submitting it as an original work-giving or receiving answers during tests or quizzes. It is your responsibility to secure your papers, so other students will not have the opportunity to copy from you or the temptation to do so-taking credit for group work when you have not contributed an equal or appropriate share toward the final result-accessing a test or quiz for the purpose of determining the questions in advance of its administration.-Using summaries/commentaries (Cliffs Notes, Spark Notes, etc.) in lieu of reading the assigned materials.-Securing test questions from students who have already completed the assessment.Alternatives to cheating and plagiarism:.No student needs to cheat or plagiarize. Toms River High School East provides a number of support services for students to help them achieve success honorably. Students who work diligently and seek appropriate help when they need it will not need to cheat or plagiarize.The following behaviors promote true student achievement:1. Be prepared. Try to keep to a realistic schedule balancing academic obligations and your social and personal life.2. Make certain that you understand your assignments and the grading assessment that will be used. If you havequestions about an assignment or an assessment, talk to your instructor. Do not relysolely upon a classmate for clarification.3. If you study for a test with a classmate, make sure that you do not sit near each other during the test since yourresponses (and errors) may be similar.4. Do not read or scan someone else’s paper before writing your own. Some of the ideas in the other person’s papermay be ideas that you would have used, but you will now need to credit the person whose paper you read for thoseideas.5. Assignments should be considered individual unless the instructor states otherwise.6. If, for whatever reason, you choose to use another’s ideas or solutions, cite that person as a source on your paper or project.7. Know what constitutes cheating, including all the variations of plagiarism.8. Use all avenues of support available to you. For help needed beyond the classroom, see your instructor, otherinstructors in the department, a peer tutor, or a parent or other adult who is well versed in the subject.9. Consult with your guidance counselor for advice on how to sustain or improve your academic performance.10. Take Advantage of the NHS tutors who offer assistance/tutoring after school in our Media Center.11. Be organized. Having class notes in an orderly, easily accessible format will save time and anxiety when studyingfor a test or writing a paper.12. Keep current with assignments. If you need to read an entire novel the evening before a test or before a paper isdue on that novel, you’re per- formance on either will suffer, and you may be tempted to find an inappropriate solutionto the problem created by time pressure.The role of parental support in a chil d’s achievement and ethical development:Parental support of academic achievement and ethical development is fundamental to students’ long-term success. The following behaviors will assist parents in promoting true student achievement:1. Teachers are available for extra help after school and in some cases, before school, and resource centers are openall day for individual assistance. In addition, peer tutors are available when extra help is needed. Encourage their use.If your child is tentative about seeking assistance, contact your child’s guidance counselor to review and discussoptions2. Assess your child’s abilities realistically. Help h er/him to choose courses in which she/he will be successful and challenged without undue stress.3. Don’t push children beyond their limits with your expectations or aspirations. Many times, students make baddecisions because the pressure to excel is greater than their ability to meet the expectations.4. If you suspect your child is experiencing difficulty in a class, please contact the teacher. The sooner the problem is identified, the sooner steps can be taken to alleviate it.5. If your child is caught cheating and you are called, please remember that this is a learning experience; help yourchild to accept the consequences for his/her inappropriate actions.Repercussions1. Any student who is caught cheating or plagiarizing will receive a grade of “zero” for the academic work involved,and the parent(s) of the student(s) will be notified. When work is copied from another student,both stud ents will be penalized with a grade of “zero.” Grades of “zero” which are the result of any form ofacademic dishonesty are irrevocable. At the teacher’s discretion, students may be required tocomplete the affected assignment- even though credit will not be awarded.. Cheating and/or plagiarism will havean adverse effect upon-application to and/or membership in the National Honor Society.4. At the discretion of the principal, students who plagiarize may be removed from certain courses. Students may be disciplined accordingly.(BASED UPON THE POLICY FROM MANCHESTER ESSEX REGIONAL HIGH SCHOOL. USED WITH PERMISSION OF TIM AVERILL.)。

1 Introduction to the Galois Theory of Linear Differential Equations

∂ ∂ , . . . , ∂x }) = inﬁnitely diﬀerExamples 1.2.2 1) (C ∞ (Rm ), ∆ = { ∂x 1 m m entiable functions on R . ∂ ∂ 2) (C(x1 , . . . , xm ), ∆ = { ∂x , . . . , ∂x }) = ﬁeld of rational functions 1 m ∂ 3) (C[[x]], ∂x ) = ring of formal power series 1 ] C((x)) = quotient ﬁeld of C[[x]] = C[[x]][ x ∂ 4) (C{{x}}, ∂x ) = ring of germs of convergent series 1 C({x}) = quotient ﬁeld of C{{x}} = C{{x}}[ x ] ∂ ∂ 5) (MO , ∆ = { ∂x , . . . , ∂x }) = ﬁeld of functions meromorphic on 1 m open,connected m O ⊂C
1.2 What is a Linear Diﬀerential Equation? I will develop an algebraic setting for the study of linear diﬀerential equations. Although there are many interesting questions concerning diﬀerential equations in characteristic p [MvdP03a, MvdP03b, dP95, dP96], we will restrict ourselves throughout this paper, without further mention, to ﬁelds of characteristic 0. I begin with some basic deﬁnitions.

Balanced Superprojective Varieties

DISTA-2007Balanced Superprojective VarietiesR.Catenacci ®,H ,M.Debernardi ®,P.A.Grassi ®,H ,K ,and D.Matessi ®®DISTA,Universit`a del Piemonte Orientale,Via Bellini 25/G,Alessandria,I-15100,Italy,H INFN -Sezione di Torino-Gruppo collegato di Alessandria K Centro Studi e Ricerche E.Fermi,Compendio Viminale,I-00184,Roma,Italy.Abstract We ﬁrst review the deﬁnition of superprojective spaces from the functor-of-points perspective.We derive the relation between superprojective spaces and supercosets in the framework of the theory of sheaves.As an application of the geometry of superprojective spaces,we extend Donaldson’s deﬁnition of balanced manifolds to su-permanifolds and we derive the new conditions of a balanced supermanifold.We apply the construction to superpoints viewed as submanifolds of superprojective spaces.We conclude with a list of open issues and interesting problems that can be addressed inthe present context.catenacc@mfn.unipmn.it,marcod@mfn.unipmn.i,pgrassi@cern.ch,and matessi@mfn.unipmn.it a r X i v :0707.4246v 1 [m a t h -p h ] 28 J u l 20071IntroductionSupermanifolds are rather well-known in supersymmetric theories and in string theory.They provide a very natural ground to understand the supersymmetry and supergravity from a geometric point of view.Indeed,a supermanifold contains the anticommuting coordinates which are needed to construct the superﬁelds whose natural environment is the graded algebras[2,3].However,the best way to understand the supermanifold is using the theory of sheaves[3,7].In the present notes we review this approach and its usefulness in theoretical physics and in particular in the last developments(twistor string theory[10]and pure spinor string theory[11]).In the case of twistor string theory,the target space is indeed the supermanifold CP(3|4) which can be described in two ways:as a supercoset of the supergroup P SU(4|4)/SU(3|4) or as a quotient of the quadratic hypersurface in the superspace C(4|4)given byα˙α|Zα˙α|2+A¯ψAψA=1(1.1)where(Zα˙α,ψA)are the supertwistor coordinates.Obviously,this equation needs a clariﬁ-cation:the commuting coordinates Zα˙αcannot be numbers for the above equation to have a non-trivial meaning.One way to interpret the above equation is using the sheaf point of view where Zα˙α,ψA are the generators of a sheaf of supercommuting algebra over open sets on CP3.In this way,the supermanifold can be viewed as(CP3,O CP3(Zα˙α,ψA))(1.2) and the equation(1.1)makes sense(see also[8]).The second way is using the functor of point.This is a functor between the category of sets and the category of supermanifolds and, as is well explained in[5]and the forthcoming sections,it assigns a point in a supermanifold in terms of a set of coordinates.The easiest way to realize the functor of point is to map a superspace into a supermanifold and describe the latter in terms of points identiﬁed by morphisms.Concretely,this amount to choose a graded algebra with N generators and represent the generators of the sheaf O CP3(Zα˙α,ψA)in terms of them.Then inserting this decomposition in(1.1),one gets a set of numerical equations for the coeﬃcients of the decomposition and they can be solved or studied by the conventional means of algebraic geometry.Of course the hypersurface(1.1)is one example of manifold that can be realized in terms of the generators of O CP3(Zα˙α,ψA)and that can be studied by means of the functor of points.Notice that also from the supercoset point of view,the technique of the functor of point gives us a representation of the supercoset in terms of the generators of a sheaf. Indeed,by multiplying supermatrices(whose entries are the generators of the sheaf)one ﬁnds that the entries cannot be numbers and they have to be promoted to the generator of a sheaf.Therefore the multiplication between matrices and the group multiplication of a supergroup has to be understood as a morphism of a ringed space.This point of view has been emphasized by Manin[3]and recently by[5,6].We provide here a more elementary explanation of the role of functor-of-point in the case of supergroup and supercosets.The purpose of this is to use the functor-of-point to deﬁne the superprojective spaces(as CP(3|4)above)and to prove the isomorphism with the supercoset point-of-view as in the purely bosonic case.In the second part of the paper,we develop two applications for superprojective spaces. Following the recent analysis of Donaldson[13]on balanced manifold,we extend his deﬁnition to supermanifolds.One ingredient is the deﬁnition of balanced submanifold of a projective space(for example a point or a line).For that we extend the integral equation given in[13] to an integration on the supermanifold.The deﬁnition of the integral of a superform in a supermanifold is not an obvious extension since a regularization is needed.This can be done using the projection forms as illustrated in[17]and discussed in more detail in[18].We brieﬂy discuss this point in the text,but we refer to a forthcoming publication for a more detailed account[21].After discussing the general theory,we provided a simple example of the embedding of P1|2into the superprojective space P2m−1|2m of sections H0(P1|2,L⊗m)where L⊗m is the m-power of a line bundle L over P1|2.In this case both the base manifold and the sections P2m−1|2m are super-Calabi-Yau spaces(in the sense that they are super-K¨a hler spaces with vanishing Ricci tensor and an holomorphic top formΩCY)and for those there is a natural measure for integrating superforms provided byΩCY∧¯ΩCY.It is shown that there are two types of conditions emerging from the extension of the Donaldson equations to the supermanifold case and therefore this restricts the number of supermanifolds that can be balanced subvarieties of superprojective spaces.In generalizing the analysis of Donaldson we have taken into account the extension of the Kodaira embedding theorem discussed in [19].The second application is to consider a set of points C0|N immersed in the superprojective space P1|N as a subvariety.In this case we computed explicitly the general expression for the case C0|2embedded into P1|2and we found the condition for the balancing of a point. We found also how the supermanifold case generalizes the classical embedding condition and we argued how one can recover the classical balancing in addition to the requirements on the parameter of the superembeddings.We showed that this is tied to the choice of the integration measure for superforms.A concluding remark:we have not explored all possible implications of our extension neither we have discussed the relation with the stability of points in the sense of Geometric Invariant Theory(GIT)[12]Nevertheless we have found rather interesting that some appli-cations admit a non trivial generalization of the usual geometric setting.These results open new questions about the geometry of sheaves and their functor-of-point interpretation.The paper is organized as follows:in sec.2we deﬁne the supermanifolds form a sheaf theory point of view.We discuss the basic architecture and the set of morphisms.In sec.3 we deﬁne superprojective spaces and in sec.4we provide a functor-of-point interpretation Part of this material is summary of notes[5].This allows us to use the local coordinates and to deﬁne the concept of a point in a supermanifold.In sec.5,we study supergroups and superdeterminant(Berezinians)from the functor-of-point perspective needed to see the deﬁnition of superprojective space as supercosets of supergroups discussed in sec. 5.1.In sec.6,we extend the construction of Donaldson to supermanifold and we deﬁne balanced supermanifolds.Finally,in sec. 6.2we discuss the balancing of points in superprojective spaces.2Supermanifolds 2.1DeﬁnitionsA super-commutative ring is a Z 2-graded ring A =A 0⊕A 1such that if i,j ∈Z 2,then a i a j ∈A i +j and a i a j =(−1)i +j a j a i ,where a k ∈A k .Elements in A 0(resp.A 1)are called even (resp.odd ).A super-space is a super-ringed space such that the stalks are local super-commutative rings (Manin-Varadarajan).Since the odd elements are nilpotent,this reduces to require that the even component reduces to a local commutative ring.A super-domain U p |q is the super-ringed space U p ,C ∞p |q ,where U p ⊆R p is open and C ∞p |q is the sheaf of super-commutative rings given by:V →C ∞(V ) θ1,θ2,...,θq ,(2.1)where V ⊆U p is and θ1,θ2,...,θq are generators of a Grassmann algebra.The grading is the natural grading in even and odd elements.The notation is taken from [4]and from the notes [5].Every element of C ∞p |q (V )may be written as I f I θI ,where I is a multi-index.A super-manifold of dimension p |q is a super-ringed space locally isomorphic,as a ringed space,to R p |q .The coordinates x i of R p are called the even coordinates (or bosonic),while the coor-dinates θj are called the odd coordinates (or fermionic).We will denote by (M,O M )the su-permanifold whose underlying topological space is M and whose sheaf of super-commutative rings is O M .To a section s of O M on an open set containing x one may associate the value of s in x as the unique real number s ∼(x )such that s −s ∼(x )is not invertible on every neighborhood of x .The sheaf of algebras O ∼,whose sections are the functions s ∼,deﬁnes the structure of a diﬀerentiable manifold on M ,called the reduced manifold and denoted M ∼.2.2Morphisms.In order to understand the structure of supermanifolds it is useful to study their morphisms.Here we describe how a morphism of supermanifolds looks like locally.A morphism ψfrom (X,O X )to (Y,O Y )is given by a smooth map map ψ∼from X ∼to Y ∼together with a sheaf map:ψ∗V :O Y (V )−→O X (ψ−1(V )),(2.2)where V is open in Y .The homomorphisms ψ∗V must commute with the restrictions andthey must be compatible with the super-ring structure.Moreover they satisfyψ∗V (s )∼=s ∼◦ψ∼.We illustrate this with an example taken from [5].Given M =R 1|2,we describe a morphism ψof M into itself such that ψ∼is the identity.Let ψ∗be the pull-back map deﬁned previously.We denote {t,θ1,θ2}the coordinates on M ,where t can be interpretedboth as the coordinate on M∼=R or as an even section of the sheaf.Since the sheaf map must be compatible with the Z2−grading,ψ∗t is an even section and(ψ∗t)∼=t.Then,ψ∗(t)=t+f(t)θ1θ2.Similarly,ψ∗(θj)=g j(t)θ1+h j(t)θ2.It is important to observe that this deﬁnes uniquelyψ∗for sections of the forma+b1θ1+b2θ2.where a,b1and b2are polynomials in t.It is therefore reasonable to expect thatψ∗is uniquely deﬁned.Let us take,for simplicity,the case whereψ∗(t)=t+θ1θ2,andψ∗(θj)=θj.(2.3)If g is a smooth function of t on an open set U⊆R,we want to deﬁneψ∗U (g).Let us expand g(t+θ1θ2)as a formal Taylor series:g(t+θ1θ2)=g(t)+g (t)θ1θ2. The series does not continue because(θ1θ2)2=0.Then,we deﬁneψ∗U(g)=g(t)+g (t)θ1θ2.Ifg=g0+g1θ1+g2θ2+g12θ1θ2, then we must deﬁneψ∗U (g)=ψ∗U(g0)+ψ∗U(g1)θ1+ψ∗U(g2)θ2+ψ∗U(g12)θ1θ2.where we have used(2.3).The family(ψ∗U )then deﬁnes a morphism between R1|2and itself.This method can be extended to the general case.Let us recall some fundamental local properties of morphisms.A morphismψbetween two super-domains U p|q and V r|s is given by a smooth mapψ∼:U→V and a homomorphism of super-algebrasψ∗:C∞r|s(V)→C∞p|q(U).It must satisfy the following properties:•If t=(t1,...,t r)are coordinates on V r,each component t j can also be interpreted as a section of C∞r|s(V).If f i=ψ∗(t i),then f i is an even element of the algebra C∞p|q(U).•The smooth mapψ∼:U→V must beψ∼=(f∼1,...,f∼r),where the f∼iare the valuesof the even elements above.•Ifθj is a generator of C∞r|s(V),then g j=ψ∗(θj)is an odd element of the algebra C∞p|q(U).The following fundamental theorem(see for example[5])gives a local characterization of morphisms:Theorem1[Structure of morphisms]Supposeφ:U→V is a smooth map and f i,g j, with i=1,...,r,j=1,...,s,are given elements of C∞p|q(U),with f i even,g j odd andsatisfyingφ=(f∼1,...,f∼r).Then there exists a unique morphismψ:U p|q→V r|s withψ∼=φandψ∗(t i)=f i andψ∗(θj)=g j.Remark.If V is a vector bundle over a smooth manifold M,then we can form its exterior bundle E=Λmax V.Let O(E)be the sheaf of sections of E.Then,locally on M,the sheaf is isomorphic to U p|q where p=dim(M)and q=rank(V).This is clearly true whenever V is restricted to some open subset of M over which it is trivial.Consequently, (M,O(E))is a super-manifold,denoted by E .Every super-manifold is locally isomorphic to a super-manifold of the form E .However we should note the important fact that E ,as a supermanifold,has many more morphisms than the corresponding exterior bundle E,because of the possibility that the even and odd coordinates can be mixed under transformations. This is well illustrated by the previous simple example.Another way to say the same thing is that there are less morphisms which preserve the bundle structure than morphisms which preserve the super-manifold structure.2.3Local charts on supermanifoldsWe describe how supermanifolds can be constructed by patching local charts.Let X=iX ibe a topological space,with{X i}open,and let O i be a sheaf of rings on X i,for each i. We write(see[4])X ij=X i∩X j,X ijk=X i∩X j∩X k,and so on.We now introduce isomorphisms of sheaves which represent the“coordinate changes”on our super-manifold. They allow us to glue the single pieces to get theﬁnal supermanifold.Letf ij:X ji,O j|Xji−→X ij,O i|Xijbe an isomorphisms of sheaves withf∼ij=Id.This means that these maps represent diﬀerentiable coordinate changes on the underlying manifold.To say that we glue the ringed spaces(X i,O i)through the f ij means that we are con-structing a sheaf of rings O on X and for each i a sheaf isomorphismf i:(X i,O|Xi)−→(X i,O i),f∼i =Id Xisuch thatf ij=f i f−1j,for all i and j.The following usual cocycle conditions are necessary and suﬃcient for the existence of the sheaf O:i.f ii=Id on O i;ii.f ij f ji=Id on O i|Xi;iii.f ij f jk f ki=Id on O i|Xijk.3Projective superspacesDue to their importance in physical applications we now give a detailed description of projec-tive superspaces.One can work either on R or on C,but we choose to stay on C.Let X be the complex projective space of dimension n.The super-projective space will be called Y.The homogeneous coordinates are{z i}.Let us consider the underlying topological space as X, and let us construct the sheaf of super-commutative rings on it.For any open subset V⊆X we denote by V its preimage in C n+1\{0}.Then,let us deﬁne A(V )=H(V )[θ1,θ2,...,θq], where H(V )is the algebra of holomorphic functions on V and{θ1,θ2,...,θq}are the odd generators of a Grassmann algebra.C∗acts on this super-algebra by:t:I f I(z)θI−→It−|I|f It−1zθI.(3.1)The super-projective space has a ring over V given by:O Y(V)=A(V )C∗which is the subalgebra of elements invariant by this action.This is the formal deﬁnition of a projective superspace(see for example[5]),however we would like to construct the same space more explicitly from gluing diﬀerent superdomains as in sec.2.3.Let X i be the open set where the coordinate z i does not vanish.Then the super-commutative ring O Y(X i)is generated by elements of the typef0z0z i,...,z i−1z i,z i+1z i,...,z nz i,f rz0z i,...,z i−1z i,z i+1z i,...,z nz iθrz i,r=1,...,q.In fact,to be invariant with respect to the action of C∗,the functions f I in equation(3.1) must be homogeneous of degree−|I|.Then,it is obvious that the only coordinate we can divide by,on X i,is z i:all functions f I are of degree−|I|and holomorphic on X i.If we put,on X i,for l=i,Ξ(i)l =z lz iandΘ(i)r=θrz i,then O Y(X i)is generated,as a super-commutativering,by the objects of the formF(i) 0Ξ(i),Ξ(i)1,...,Ξ(i)i−1,Ξ(i)i+1,...,Ξ(i)n,F(i)aΞ(i),Ξ(i)1,...,Ξ(i)i−1,Ξ(i)i+1,...,Ξ(i)nΘ(i)a,where F(i)0and the F(i)a’s are analytic functions on C n.In order to avoid confusion we haveput the index i in parenthesis:it just denotes the fact that we are deﬁning objects over thelocal chart X i.In the following,for convenience in the notation,we also adopt the conventionthatΞ(i)i =1for all i.To explain the“coordinate change”morphisms let us recall what happens in the ordinary complex projective spaces.If we consider P n(C)with the ordinary complex analytic structure,then,over the aﬃneopen set X i where z i=0,we can deﬁne the aﬃne coordinates w(i)a=z ai ,a=i.The sheaf ofrings over X i is H(X i),the ring of analytic functions over X i.Every element f of H(X i)can also be expressed as a function in homogeneous coordinates F(z0,z1,...,z n).Two functions, F(i)on X i and F(j)on X j,represent“the same function”on the intersection X i∩X j if,when expressed in homogeneous coordinates,they give the same function F.The isomorphismbetween(X i∩X j,H(X i)|Xj )and(X j∩X i,H(X j)|Xi)sends F(i)to F(j),i.e.expresses F(i)with respect to the aﬃne coordinates w(j)a=z az j .The total manifold is obtained by gluingthese domains X i as in the previous section.We now return to considering the super-projective spaces.We have the two sheavesO Y(X i)|Xj and O Y(X j)|Xi.In the same way as before,we have the morphisms given bythe“coordinate changes”.So,on X i∩X j,the isomorphism simply aﬃrms the equivalence between the objects of the super-commutative ring expressed either by theﬁrst system ofaﬃne coordinates,or by the second one.So for instance we have thatΞ(j)l =z lz jandΘ(j)r=θrz jcan be also expressed asΞ(j) l =Ξ(i)lΞ(i)j,Θ(j)r=Θ(i)rΞ(i)j.Which,in the language used in the previous section,means that the morphismψji gluing(X i∩X j,O Y(X i)|Xj )and(X j∩X i,O Y(X j)|Xi)is such thatψ∼jiis the usual change ofcoordinates map on projective space andψ∗ji (Ξ(j)l)=Ξ(i)lΞ(i)j,ψ∗ji(Θ(j)r)=Θ(i)rΞ(i)jThe super-manifold is obtained by observing that the coordinate changes satisfy the cocycle conditions of the previous section.4The functor of pointsWe now wish to explain how the physicists’interpretation of the z i’s as“even coordinates”and theθj’s as“odd coordinates”can be obtained from the“super-ringed space”interpreta-tion of supermanifolds through the concept of“functor of points”.The key to understanding this is Theorem1.Given two supermanifolds X and S,the S-points of X(or the points of X parametrized by S)are given by the setX(S)=Hom(S,X)={set of morphisms S→X}.X is the supermanifold we want to describe and S is the model on which we base the description of X.Changing S modiﬁes the description of X.The functor which associates Sto X(S)is a functor between the category of supermanifolds and the category of sets(which are the“points”of the supermanifolds).See also[6]for more details.Let us interpret this in the case when X=V r|s and S=U p|q.According to Theorem1, a morphismψ∈Hom(U p|q,V r|s)is uniquely determined by a choice of r even sections and s odd sections of C∞p|q(U),i.e.morphisms are in one to one correspondence with(r+s)-tuples (f1,...,f r,g1,...,g s),where f j’s are even and g j’s are odd in the algebra C∞p|q(U).If wedenote byΓ0q (U)andΓ1q(U)respectively the set of even and odd sections of C∞p|q(U),thenthe above fact is expressed asHom(U p|q,V r|s)=(Γ0q (U))r×(Γ1q(U))s.(4.1)The sub-index q denotes the“number of odd generators”of the algebra we are considering.In particular,if S=R0|q,thenHom(R0|q,V r|s)=(Γ0q )r×(Γ1q)s(4.2)where(Γ0q )and(Γ1q)represent the even and the odd component of a Grassmann algebra withq generators,respectively.One could say that the“super-ringed space”structure of X encodes the information of how the even and odd coordinates(z,θ)glue together,but independently of the number of generators of the underlying super-algebra.The number of generators(q in the above case) can beﬁxed by taking a supermanifold S and constructing Hom(S,X).We will see some examples shortly.4.1Coordinates of Superprojective Spaces.We are going to consider the superprojective spaceP p|q=(P p,O P p)(4.3) which is deﬁned as in section3as a ringed space and dim(O P p)=q.We want to describe the set of C0|N−points of this space.The space C0|N can be viewed as the super-commutative ring O C0over the single point(denoted by P)of the corresponding topological space C0and can be identiﬁed precisely with the Grassmann algebra with N generators that we denote byΓN.Let’s consider the open subsets{X i},i=0,1,2,...,p,of P p where z i=0,with thecorresponding super-commutative ring O Xi .A morphism between C0|N and(X i,O Xi)iscompletely deﬁned by the pull-back for each generator of the ring O Xiτ(i)∈Hom(C0|N,(X i,O Xi )),τ∗(i):O Xi→O C0(4.4)where O C0=C[θ1,...,θN]=ΓN.To clarify this point,we take the generators of O Xi:Θ(i)j ,j=1,...,q and the aﬃne coordinatesΞ(i)jon X i and we map into C[θ1,...,θN]asfollowsτ∗(i)(Ξ(i)j)=f(i)jj=1,...,p(4.5)τ∗(i)(Θ(i)r)=η(i)rr=1,...,qwhere the f(i)j(resp.η(i)r)are even(resp.odd)elements of the Grassmann algebraΓN.It isclear thatτ∼(i)(P)=((f(i)1)∼,...,(f(i)p)∼).We therefore see that for every i,Hom(C0|N,(X i,O Xi))can be identiﬁed with a copy of(Γ0N )p×(Γ1N)q.To obtain all the possible morphisms from C0|N to P p|q,we must take into account that the latter is built by“gluing”super-domains by means of the“coordinate change isomorphisms”,this corresponds to gluing together all copies of(Γ0N )p×(Γ1N)q for all possible i’s.Since amorphism in Hom(C0|N,P p|q)must be compatible with the restriction maps,it must commute with the“coordinate changes”.This means that,ifτ∗(j)is the pull-back of a morphism toX j,andψ∗ij :O Xj|Xi∩X j−→O Xi|Xi∩X jis the isomorphism which represents“coordinatechanges”,thenτ∗(j)=τ∗(i)◦ψ∗ij.This then induces a map between subsets of the i-th and j-th copy of(Γ0N )p×(Γ1N)q asfollows(f(i)1,...,f(i)p)→(f(i)j)−1(f(i)1,...,1,...,f(i)p),(η(i)1,...,η(i)q)→(f(i)j)−1(η(i)1,...,η(i)q).By means of this map we glue the two copies together.Performing all these gluings gives a model for Hom(C0|N,P p|q),consisting of the C0|N-points of P p|q.Another way to interpret this model is as follows.We consider a set of“homogeneous”(even and odd)generators z0,...,z p,θ1,...,θq,where the z j’s are inΓ0N and at least one themis invertible and theθj’s are inΓ1N .One obtains the local generators on each X i simply“dividing”by z i(exactly like in the standard projective case,when one looks for the“aﬃne coordinates”).This way we see that we can identifyHom(C0|N,P p|q)=(Γ0N)p+1\B p+1×(Γ1N)q(Γ0N)∗,where(Γ0N )∗is the set of the even invertible elements and B=(Γ0N)\(Γ0N)∗.This model isexactly the generalization of the projective space as a supermanifold in the sense of Rogers, Bruzzo and others(see book[7]for a complete discussion).5Supergroups and SuperdeterminantsAs another illustration of the meaning of the functor of points we consider the case of supergroups.For simplicity we will just look at the cases of GL(1|1),SL(1|1)andﬁ-nally we will give another construction of the superprojective space as the quotient space SU(n|m)/U(n−1|m).Let us now consider the simplest case of supergroup GL(1|1).As a supermanifold, GL(1|1)is isomorphic to the super-domain U2|2=(U2,C∞2|2),where U2=(C∗)2.If(z1,z2) are the coordinates on U2andθ1,θ2are the generators of the Grassmann algebra,it isconvenient to use the notation in matrix formz1θ1θ2z2.(5.1)We can deﬁne the“product”on GL(1|1)as a morphismψ∈Hom(GL(1|1)×GL(1|1),GL(1|1))such thatψ∼:GL(1|1)0×GL(1|1)0−→GL(1|1)0(z1,z2)×(z3,z4)−→(z1z3,z2z4),ψ∗:C∞(U2)[ϑ1,ϑ2]−→C∞(U2×U2)[θ1,θ2,θ3,θ4]w1ϑ1ϑ2w2−→z1z3+θ1θ4θ1z4+z1θ3θ2z3+z2θ4z2z4+θ2θ3,where the action of the pull-back morphismψ∗has been speciﬁed only for the generators of the algebra(see Theorem1).We now apply the functor of points to recover the usual interpretation of GL(1|1)as the set of“invertible supermatrices”.Take as model space S=C0|q,then Hom(S,GL(1|1))canbe identiﬁed with the set of matricesg=ψ∗z1ψ∗θ1ψ∗θ2ψ∗z2,(5.2)whereψ∗z i are even elements of the Grassmann algebraΓq,whose value is diﬀerent from zero,andψ∗θi are odd elements.To simplify notation we denoteψ∗z i(resp.ψ∗θi)by z i (resp.θi).The above“product”becomes the usual multiplication of super-matrices as follows.A morphism form S to GL(1|1)×GL(1|1)is given by a pair of matrices,g1and g2,as above. Composition with the product morphism gives a morphism from S to GL(1|1),represented by a matrix g3,which can be seen to be given by the usual multiplication of matrices:g1=z1θ1θ2z2,g2=z3θ3θ4z4(5.3)g3=g1g2=z1z3+θ1θ4θ1z4+z1θ3θ2z3+z2θ4z2z4+θ2θ3.Recall the classical formula for the superderminant(or Berezinian)of a super-matrix in GL(1|1):sdet(g)=Ber(g)=z1z21+θ1θ2z1z2(5.4)which is well deﬁned if z2=0.The Berezinian can also be understood from the sheaf point of view,as a morphism Ber from GL(1|1)to C1|0:Ber∼:GL(1|1)0−→C (z1,z2)−→z1/z2,Ber∗:C∞1|0−→C∞2|2w−→z121+θ1θ212.Next,we consider a subset of supermatrices GL(1|1)with the property that“the su-perdeterminant is1”.They are denoted by SL(1|1).We want to describe this space using the sheaf theoretic interpretation of supermanifolds,by restricting the base manifold and considering an appropriate quotient sheaf.We need to give a meaningful interpretation ofthe relationz1z21+θ1θ2z1z2=1.(5.5)We do it as follows.Let J be the ideal in C∞2|2generated byz1 z21+θ1θ2z1z2−1.(5.6)The base manifold of SL(1|1)is the support of this ideal,i.e.the subset X⊂C∗×C∗of points around which no element of J is invertible.Clearly X is the diagonal in C∗×C∗,i.e. the set where z1=z2.The sheaf O X is the restriction to X of the quotient sheafC∞2|2/J.(5.7) It remains to show that this ringed manifold SL(1|1)is really a supermanifold,i.e.it is obtained by pasting super-domains.In fact observe that the relation(5.6)tells us that overan open set V⊂X,z1=z2−θ1θ2z2.Therefore the ring over V is C∞(z2−θ1θ2z2,z2)[θ1,θ2]which can be seen to be isomorphic to a super-ring of the type C∞(z)[Ψ1,Ψ2],so locally SL(1|1)is isomorphic to a superdomain.What we have done here is to show explicitly that SL(1|1)is a sub-supermanifold of dimension1|2of GL(1|1)in the sense of[5].To conclude the description of X=SL(1|1),we present its interpretation by means of the functor of points,using the model space S=C0|N.Then,the morphisms in Hom(S,X) can be viewed as the morphisms in Hom(S,GL(1|1))such that:i.)the map between the underlying topological spaces has image contained in the diagonalof C∗×C∗;ii.)the pull-back map descends to the quotient,i.e.ψ∗(j)=0for any j∈J.Then,the set Hom(S,X)can be viewed as the matrices of the formg=ψ∗z1ψ∗θ1ψ∗θ2ψ∗z2,(5.8)with the conditions that:1)the value ofψ∗z1is equal to the value ofψ∗z2(by(i)),and2) the super-determinant of the matrix in(5.8)is1(by(ii)).We start describing GL(1|1,C)in a diﬀerent way,by passing to real super-groups.We use the following idea:think about C n,with a complex basis{v1;v2;...;v n}.Then,C n can be viewed as a real2n-dimensional space,with a real basis{v1;iv1;v2;iv2;...;v n;iv n}.。

商业道德论文

housing and in share prices,the massive loss of jobs,and the slowing of economic growth.It thus includes the global recession and sovereign debt crisis that continue,and threaten to lead to stagﬂation.This short article has three objectives.First,it will outline what constitutes ethical conduct by business.Second,it will suggest why ﬁnance seems so often to be associated with unethical conduct.And third,it will identify what conduct leading to and responding to the Global Financial Crisis was indeed morally wrong.The conceptual and philosophical approach of this article is thus substantially different from most responses to the crisis,whose focus has instead been macroeconomic analysis,institutional design 2,and/or public policy,especially deregulation.3Even commentators ostensibly dealing with ethics have characteristically just condemned greed (e.g.Tett 2009;Lewis et al.2010),or deplored the absence of integrity.4This article,in contrast,attempts to offer an analytical framework for identifying exactly what was unethical,and why it was.2.Business ethics properly understoodTo understand what constitutes ethical conduct by business,one must ﬁrst understand what business is.Properly identifying the purpose of business is vital,because the values of business ethics are just those that must be respected for the business purpose to be possible.The speciﬁc objective that is unique to business,and that distinguishes business from everything else,is maximising owner (ﬁnancial)value over the long term by selling goods or services 5.Long-term views require operating over time,and thus conﬁdence in a future.Conﬁdence requires trust,so the conditions of trust must be observed:lying,cheating and stealing are therefore ruled out.Equally,owner value presupposes ownership and therefore respect for property rights.In order not to be ultimately self-defeating,business must be conducted with honesty and fairness,and without physical violence or coercion.Collectively,these constraints embody what may be called ‘Ordinary Decency ’.Furthermore,business that is directed at achieving its deﬁnitive purpose encouragescontributions to that purpose,and not to some other;classical ‘Distributive Justice ’is alsoessential.Just as Ordinary Decency is distinct from vague notions of ‘niceness’,this concept of justice has nothing to do with modern attempts to redistribute wealth on ideological grounds.What Distributive Justice requires is simply that within an organisation,contributions to the organisational objective be the basis for distributing organisational rewards.Though the term ‘Distributive Justice’may be unfamiliar,the underlying concept is widely recognised.It is implicit in the commonly accepted view that productive workers deserve more than shirkers;when properly structured,both performance-related pay and promotion on merit areexpressions of Distributive Justice.The key to Realist business ethics is very simple:business is ethical when it maximiseslong-term owner value subject to Distributive Justice and Ordinary Decency.If an organisation is not directed at maximising long-term owner value,it is not a business;if it does not pursue that deﬁnitive business purpose with Distributive Justice and Ordinary Decency,it is not ethical.3.The prominence of ﬁnancial problemsIf business ethics is so straightforward,why do ﬁnancial matters seem to present so many ethical problems?19economic affairs volume 33,number 120 e.sternbergOne key reason is the sheer pervasiveness ofﬁnance.Since all business dealings involveﬁnance,so do most business problems–including business’s ethical problems.Finance is not exceptionally problematical,but it is,in business,virtually inescapable;everything a business does hasﬁnancial ramiﬁcations.Finance also suffers from many of the same features that make business itself seem ethically suspect.It is the locus of decisions that can involve vast sums and equally strong temptations,and its practitioners are often presumed to be clever manipulators of money and of men.And sadly,some prominent examples ofﬁnancial loss have indeed been the consequence of moral wrongdoing,of deceit and outright theft.6Even in such cases, however,the misconduct is typically reﬂected inﬁnance;it is not intrinsic toﬁnance.Major business scandals tend to be associated withﬁnance,because they are associated with ﬁnancial losses.But it would be wrong to conclude either thatﬁnancial wrongdoing is the cause of those losses,or thatﬁnance is especially unethical.Businesses–including banks–typically lose money because their products or their marketing are misjudged,or because their operations or their staff are badly managed.Such failures always haveﬁnancial consequences, and are ordinarily expressed inﬁnancial terms,but they are not themselvesﬁnancial shortcomings.Even whenﬁnancial problems are at fault,and businesses suffer from bad debts or mismatched funding or undercapitalisation,the failings involved are more often the result of folly than of fraud.Indeed,unethical conduct inﬁnance may be as much a response toﬁnancial losses as a source of them;it is when businesses are weak that the temptation to falsify results may be the greatest.4.Bank conduct and the GFCWhat does this Realist understanding of ethical conduct by business,and of the place of ethics inﬁnance,indicate aboutﬁnancial institutions?In what ways,if any,did they act unethically, and contribute to the GFC?Recall that according to conventional assessments,they were evil.According to one academic report,During the second half of2008...all major world markets...were devastated by the aftermath of unethical lending practices by major lending institutions....Prudence and ethics were pushed aside as greed overcame good judgment among mortgage lenders nationwide.(Lewis et al.2010,p.77)What exactly is it that they did wrong?4.1.GreedTheﬁrst criticism that needs to be challenged is the suggestion that greed dominated all,and that its very presence rendered the associated acts immoral.Both claims are false.Greed is a motive,which undoubtedly did impel someﬁnanciers,and prompt some unethical conduct.But(even)ﬁnanciers are motivated by all sorts of impulses other than greed:by peer pressure and pride,by fashion and laziness,by curiosity and by sincere wishes to exercise their creative talents and solve problems.Greed is in fact responsible for far fewer business actions,and a fortiori for fewer business evils,than is commonly supposed.Moreover,even to the extent that greed is the operative motive,it is a relatively clean one: cupidity is,in an important sense,self-correcting.Those who genuinely want to acquire andkeep wealth will avoid conduct,including unethical conduct,that reduces wealth;the amounts of money the greedy seek may be unlimited,but the ways of achieving wealth are not.Finally,and still more fundamentally,even when greed is the motive,it has no necessaryeffect on the ethics of the actions it induces:the moral status of an act is largely independent of the motive that prompts it.4.2.Acts vs.motivesA motive is that which induces someone to act.It often refers to the personal,usuallyemotional,satisfactions that a person may seek in pursuing the objectives that deﬁne activities 7;a motive can also characterise the way in which deﬁnitive objectives are pursued.That which is done ,however,the act that is actually achieved or accomplished,can usually be abstracted away from the doer’s motive and evaluated separately.The same morally good act can be done from all sorts of motives.A person can,forexample,save a child from drowning out of a love of children,or fear of being called a coward.He can do it because it is his job,or because it is his moral duty.He can do it out of hope for publicity or a spirit of protectiveness 8.He can even do it to spite the child’s murderous parents or to upset his rival lifeguards.Whatever the motive,however,a good act –that of saving a child’s life –has been performed.That would be true even had the lifesaver been wicked and his motives thoroughly vicious:had he pulled the ﬂoundering child out of the sea only because he wanted to kidnap it,the child would still have been saved from drowning.Similarly,inbusiness,morally correct acts can be undertaken for all sorts of reasons.A ﬁrm can offer equal pay for equal work in order to champion justice or undermine the unions or simply obey the law.The moral rightness of acts is perfectly compatible with the full range of motives –prudence and ruthless selﬁshness as well as duty and altruism.Similarly,all sorts of motives,even noble ones,can give rise to immoral acts;worthyintentions are no proof against ignorance or error or foolishness.Genuinely devoted to justice,and with the best will in the world,a jury may nonetheless send the wrong man to gaol and a manager may promote the wrong person.The moral quality of an act is not determined by the motive that inspires it.Acts motivated by greed are not necessarily immoral.4.3.Subprime mortgagesBut perhaps the problem highlighted by the Global Financial Crisis is that there is something wrong about particular ﬁnancial instruments.Consider subprime mortgages.One of the difﬁculties that hinders the proper assessment of subprime mortgages,is ageneral lack of clarity as to what they are.‘Subprime mortgages’are more accurately called ‘subprime mortgage loans’.They are actually a combination of two things:a loan to purchase real estate,and a mortgage,or conditional conveyance of property as collateral for the loan.Colloquially,however,they are regarded as a unit,and referred to as ‘subprime mortgages’.There is,however,no generally accepted deﬁnition of what makes a mortgage loan‘subprime’.The (US)Federal Reserve Board,Federal Deposit Insurance Corporation (‘FDIC’),Ofﬁce of Thrift Supervision,and Ofﬁce of the Comptroller of the Currency (‘OCC’)generally categorise as ‘subprime’mortgages that are executed by borrowers whose FICO (Fair Isaac Credit Organization)credit scores are less than 660(OCC 2001,p.3);the US Department of21economic affairs volume 33,number 122 e.sternbergHousing and Urban Development(‘HUD’)and(sometimes)the OCC apply the speciﬁc designation‘subprime’only to the subset of those loans for which FICO scores are less than 620(OCC2009,p.11;HUD).The Federal National Mortgage Administration(‘FNMA’,‘Fannie Mae’),in contrast,classiﬁes as subprime the mortgages it purchases from subprime originators; it is unclear what renders an originator subprime.Also often classiﬁed as‘subprime’are securities that are otherwise labelled‘Alt-A’.These are pools of mortgage loans whose terms are somehow deﬁcient,because of the mortgages’high loan-to-value ratio,interest only payments,adjustable rates,and/or insufﬁcient documentation.By mid2007,27million mortgages–c.50per cent of all US mortgages–were subprime or Alt-A(Wallison2011,p.451).What is not disputed about subprime mortgages is that they areﬁnancial instruments.As such,however,subprime mortgages are not the sorts of things that can themselves have a moral status:a subprime mortgage can no more be ethical or unethical than a spoon can or a thermometer.The proper objects of moral judgement are not artefacts,but people and their actions.What can be morally assessed are the uses to which objects are put;it is those evaluations that typically characterise objects derivatively and colloquially.Artefacts that are themselves morally neutral can be used in good or bad ways,to further good and bad ends;the same candlestick that provides light can be used as an aggressive weapon.Artefacts can also be good or bad ones of their kind,that is,well or badly able to serve their purposes.A good candlestick holds a candleﬁrmly in an upright position so that it sheds light;a bad candlestick lets candles droop and drip.Moral judgements concerning subprime mortgages properly relate to their uses and their design.As business instruments,loans are ethical if they are directed at maximising long-term owner value while respecting Distributive Justice and Ordinary Decency.A loan is most likely to maximise long-term owner value for its issuer if its funding is cost-effective,and if it gets repaid in full and on time.Timely and complete repayment are,in turn,most likely if the borrowers are carefully selected.When funds are lent for long periods,they typically need the additional security of collateral to offset their higher risk.A good mortgage loan is one backed by property whose sale would be sufﬁcient to repay the lender promptly if the borrower defaulted on the interest or principal. Additional requirements for a mortgage or any other loan’s being ethical are that the methods used in generating and selling and distributing it be honest and fair,and involve no coercion.A bad loan is one that fails to satisfy any of these conceptual criteria.On what grounds,then,are subprime mortgages so often condemned?It cannot be that their inferior quality came as a surprise:like‘junk bonds’,subprime mortgages indicated their deﬁciencies in their very name.Nor can their moral status simply reﬂect their being substandard compared with ordinary,prime,mortgage loans.When subprime mortgages are properly structured,so that they are likely to get repaid proﬁtably,in full and on time(as historically most were),they can still be good business.To compensate for the greater risk of default resulting from the inferior credit quality of the borrower,or from more complex terms, a good subprime mortgage typically bears higher rates of interest than better-quality loans, and more restrictive covenants.Nor can the problem with the subprimes be their supposed complexity.Loans with an adjustable rate of interest are commonplace for other sorts of borrowing(mercial loans,and credit cards),and are standard for non-US mortgages (including those in the UK).One hundred per cent mortgages and endowment mortgages (ones on which only interest is paid throughout the life of the loan,and the principal gets repaid by the maturation of a lifeinsurance policy)have also featured in the UK ...though they have been considered suitable mainly for the highest-quality borrowers.The problem with subprime mortgages also cannot be the fact that they were pooled into securities and sold,rather than having been kept by the originating institutions.Originally initiated by a government agency,the Federal Home Loan Mortgage Corporation (‘FreddieMac’),this practice has been condemned as ‘origination for distribution’.But what’s wrong with that?Most products (e.g.shoes,computers)are created with the expectation that they will be sold rather than kept by their producers.Nor does any special problem arise because the products distributed are ﬁnancial instruments.There is a more than 40-year history ofnon-recourse securitised debt obligations based on,for example,car loans,credit card loans.They have been unproblematic,even though the vendors and purchasers of those securities equally had no responsibility for generating or monitoring the underlying,constituent loans.The reason that the subprime mortgages associated with the global ﬁnancial crisis (for simplicity,‘GFC subprimes’)were so problematical,is that they were not properly structured.They failed to meet basic standards of prudent lending,and consequently suffered much higher rates of default than subprime loans had done historically.Unlike traditional mortgages,the GFC subprime mortgages were issued to borrowers whose credit quality was signiﬁcantly lower even than that of previous subprime borrowers.In addition,the loans were typically structured without the protections that would normally have been included to compensate for less creditworthy borrowers.9‘Know your customer’is a basic banking principle.It is minimally prudent,in assessing whether large loans are likely to get repaid,to obtain basic information about the borrowers,and to demand evidence of their creditworthiness.Accordingly,loan applicants normally have to submit tax returns,personal and bank references,employment and residence histories,and other such documents,to supplement the information available from commercial credit scores and police reports.Such basic information was,however,seldom provided or even requested for recipients of GFC subprime mortgages.Indeed,so little evidence was there for many of those loans,that the borrowers were known colloquially as NINJAs:borrowers with No INcome,Jobs,or Assets.A further defect of the GFC subprimes is that the inadequacy of the borrowers’creditquality was not offset by the protections normally required in such ually,even prime borrowers are obliged to pay cash for a signiﬁcant portion of the properties they purchase.Down payments normally serve two important purposes.They protect the lenderagainst declines in the value of the collateral that would result from falling property prices.And they constitute a stake that the borrower stands to lose if he defaults on the loan.Historically,down payments of at least 20per cent of the mortgaged property’s assessed value wererequired.But for many GFC subprime mortgages,no down payment at all was needed:those loans were for 100per cent of the property’s purchase price.By making it possible for people with little or no savings to acquire property,minimal down payments made home ownership accessible to many who might not previously have had the opportunity.But by not requiring purchasers to risk any of their funds,it also strongly encouraged speculation,and helped to intensify both the property price boom and the number of defaulters.23economic affairs volume 33,number 124 e.sternbergMortgage loans that are not properly structured and/or ones that are issued to borrowers with inadequate credit quality are very risky;there is a high probability that will not get repaid. To the extent thatﬁnancial institutions made such loans,they did indeed act unethically.What was unethical,however,was that they violated the requirement to maximise long-term owner value:the mortgage lenders were insufﬁciently businesslike.Generating defective products is seldom a good strategy for maximising owner value.Even if the defective product is sold off,and even if its purchaser has no recourse to the originator, a reputation for defective products is a serious business liability.Consider the damaging effect on Toyota sales and shares of fears that some of its models might haveﬂaws.Though subprime loans were not as life-threatening as stuck accelerators,they could be–and indeed have been–detrimental to the wealth of bankers and the lives of banks.Why would lenders issue them?5.Moral hazardsA key factor was the presence of moral hazards.A moral hazard exists‘when the rules of an institution provide a positive incentive to do the wrong thing’(Sternberg1994/2000,p.103). Unfortunately,moral hazards and perverse incentives pervade the Americanﬁnancial markets as a result of government regulation.One of the most basic is deposit insurance.Because depositors know that funds up to the FDIC limit10are guaranteed by the US government,they have no incentive to entrust their savings only to prudent banks,or to monitor the operations of the banks holding their savings. In turn,the banks receiving such deposits are afforded a one-way bet.They can apply the insured funds to risky ventures in hopes of achieving high gains,conﬁdent that any losses will be made up by the US government.A second moral hazard is provided by the government’s provision of mortgage insurance. To encourage lending to less qualiﬁed borrowers,the US government has long offered guarantees of mortgages through the Federal Housing Administration(‘FHA’).Banks need not be concerned about the borrowers’creditworthiness,because the government covers the losses of defaults.By2006the FHA was the largest insurer of mortgages in the world,having guaranteed34million mortgages since its creation in1934(HUD2006).A third moral hazard is provided by the government’s guarantees of the major mortgage funding institutions.Fannie Mae and Freddie Mac raise money through the bond markets at preferred rates that reﬂect their status as Government Sponsored ing that money,Fannie and Freddie buy mortgages from lending institutions,thus supplying lenders with the funds to continue making loans.Prior to the GFC,Fannie and Freddie were ostensibly private corporations.But it was widely–and accurately11–believed that they enjoyed government guarantees.Because Fannie and Freddie expected that any losses they sustained on the mortgages they bought would be covered by the federal government,they had little incentive to restrict their purchases to properly structured mortgages,or to monitor the mortgages they acquired.Supporting housing is also behind a fourth moral hazard:tax relief on mortgage interest. This policy makes borrowing more attractive than saving,and makes real estate more attractive than other investments.It thereby distorts the market for housing and all alternative investments.As a result,both the demand for houses and the price of houses are keptartiﬁcially high.Although these moral hazards signiﬁcantly antedated the GFC,their effects wereexacerbated by a series of government regulations that further skewed lending decisions.These regulations,constituting a ﬁfth moral hazard,aimed at promoting home ownership for minorities.To that end,they obliged lenders to make loans to riskier borrowers,or suffer serious penalties.Such hazardous regulation included the following (Pinto 2010):1.The Community Reinvestment Act (‘CRA’),signed by President Carter in October 1977,outlawed redlining,the practice of refusing mortgages to everyone in a designatedgeographical area,without regard to their individual creditworthiness.Individualassessment was considered unnecessary,because it was thought that ‘low-quality housing and high levels of unemployment and welfare dependency made local residentsunattractive as borrowers’(Butler 2009,p.53).The redlined neighbourhoods were typically poor,and often black or Hispanic.2.In 1991the Home Mortgage Disclosure Act (‘HMDA’)rules were strengthened to include a speciﬁc demand for racial equality.Rules in place since 1975had already forced lenders to provide detailed reports about the identities of their borrowers;compliance wasincreasingly monitored by government-funded ‘community’groups like the Association of Community Organizations for Reform Now (‘ACORN’).3.In 1992the Federal Reserve Bank of Boston published a manual specifying that ‘criteria that would normally reduce the chances of a loan being granted ...should not affectlending decisions’(Butler 2009,p.53).Among the criteria excluded were a mortgageapplicant’s lack of credit history,using loans or gifts as their mortgage deposit (FRBB 1992,p.14),and using unemployment beneﬁts as their basis for income (FRBB 1992,p.15).4.Also in 1992,the mission of Fannie and Freddie was changed from providing liquidity to the mortgage markets,to ‘supporting affordable housing’.This political objective overrode any interest in ﬁnancial prudence;their criteria for purchasing mortgage loans wereloosened accordingly.In 1999,HUD ruled that by 2001,the percentage of mortgages that Fannie and Freddie purchased in respect of loans for affordable housing had to increase to 50per cent (Holmes 1999).In 2004,the requirement was increased:by 2008,loans to ‘low-and moderate-income families and underserved communities’had to reach 56per cent (HUD 2004).Not surprisingly,in 2008Fannie and Freddie plunged into massivedeﬁcit and collapsed ...despite being subject to 236government regulators (Butler 2009,p.55;Redwood 2008).5.In 1995,the Community Reinvestment Act regulations were amended (CRA 1995)to oblige lenders not just to avoid redlining,but alsoto ignore most of the traditional criteria of credit-worthiness in their loan decisions.Mortgagescould now be any multiple of income;a person’s saving history was irrelevant;applicants’income did not need to be veriﬁed;and participation in a credit counselling programme could be taken as proof of an applicant’s ability to manage a loan.Violation of CRA regulations could be a violation of equal opportunity laws producing exposure to actual damages plus punitive damages of$500,000....Under the CRA,if a lender wants to change its business operation in any way –merging with another bank,opening or closing branches,or developing new products –it mustconvince the regulators that it will continue to make sufﬁcient loans to the government’s preferred groups of borrowers.(Butler 2009,pp.53–4)1225economic affairs volume 33,number 126 e.sternbergThe regulations above accompanied and exacerbated the progressive relaxation of standards exercised by the government housing agencies.FHA down payment requirements,for example, plummeted.On housing amounts over$25,000,they were90per cent in1970,but a mere5per cent in1980;only theﬁrst$25,000qualiﬁed for a3per cent down payment.By1990,however,if housing cost less than$50,000,the full amount was eligible for the3per cent down payment.(Monroe2001, p.46)Not surprisingly,the FHA,Fannie and Freddie routinely guarantee over85per cent of all new US mortgages(Economist2011).Damaging though these regulations and institutions were,an even more pervasive and destructive(sixth)moral hazard has been provided by lax monetary policy.Since2000,the US Fed has kept US interest rates artiﬁcially low,and thus perverted the signals that would have been given by freely determined prices.Because low interest rates made mortgages appear inexpensive,people bought houses rather than other items.The demand for houses caused real estate prices to boom.Those inﬂated prices in turn made using houses as collateral seem safer than it was,and encouraged banks to make still more mortgage loans.Because interest rates were so low,however,banks could only maintain their loan yields by lending to ever riskier borrowers.But the seriousness of those risks was obscured,because inﬂation fuelled expectations that property prices would continue to rise.This heavily regulated environment provided overwhelming government incentives for banks to make bad loans.But disastrous though government action was in precipitating the globalﬁnancial crisis,banks were also culpable.They were complicit insofar as they succumbed to the government incentives:strong though the incentives to unethical conduct were,some banks successfully resisted.13Insofar as banks and otherﬁnancial institutions solicited or supported those regulatory constraints,they were guilty of even greater wrongdoing. Government regulation always involves coercion;as such,it automatically violates one of the deﬁning components of Ordinary Decency.‘Predatory do-gooding’is unethical even when prompted by ostensibly generous motives.It is wrong to take advantage of the less articulate, intelligent,or sophisticated to impose costs on them or others through paternalistic regulation. All government regulation14has a heavy price:independent of anyﬁnancial costs,coercive regulation limits individual liberty.Were any of the other elements of Ordinary Decency violated?It would certainly have been unethical to generate loans by lying,or by cheating prospective borrowers.Most mortgage originators did not stand in aﬁduciary relationship to their customers.But it would still have been unethical to misrepresent the costs or risks of taking on a mortgage,and wrong to pressure prospective borrowers to take out loans that they could not reasonably be expected to afford and repay.15Predatory lending,and abusive practices by lenders,are indeed unethical.But so is predatory borrowing:taking out a loan that one is not willing,able or intending to repay is a form of stealing.Similarly unethical is obtaining a loan on false pretences.But‘liar loans’were endemic.The requirement for honesty works both ways,as does caveat emptor.These moral criteria apply both to individual borrowers,and toﬁnancial institutions themselves.Financial institutions had no excuse for not understanding and properly evaluating the securitised obligations that they bought and sold.Although the chief ethical error of mostﬁnancial institutions was being insufﬁciently directed at long-term owner value16,many also violated Distributive Justice.Distributive Justice requires that compensation reﬂect contributions to the organisational goal.In a business,。

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A note on fructification of Plagiozamites Zeiller (Noeggerathiales) based on association

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慷慨援助之举的英语作文

Note on a Theorem of Munkres

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慷慨援助之举的英语作文

Note on a Theorem of Munkres

小学上册第十四次英语第1单元暑期作业

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【2018极品PPT模板】稳重大气扁平商务红黑提案汇报演讲发布

Initial Remark of France

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Balanced Superprojective Varieties

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