Blockchain Papers

Haruo Hosoya, Kikuko Hosoi, Ivan Gutman

No abstract is available for this record.

K.T. Parker

A review of optimal control theory for linear systems with quadratic cost functions is presented. Some of the theoretical and practical limitations are discussed with special reference to distributed parameter systems. First a procedure is described for finding the optimal control by constructing a sequence of controllers that converges to the optimal; this method is valid for systems of infinite dimension provided that the operators in the state differential equation satisfy certain conditions. The proof is carried out both for the finite and infinite time interval and the connection is shown with the Riccati equation. The main problem in implementation is that one needs complete knowledge of the state at all times in order to build the optimal controller, this is almost certainly impossible for distributed parameter systems. When the state cannot be measured completely it is proved that an optimal control is realisable for time invariant finite dimensional systems.
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\nThe problems of finding this control are then investigated and computational methods discussed. If the optimal control with complete knowledge of the state cannot be implemented, a method is presented whereby one can find bounds on the possible increase in the value of the cost function arising from the use of some sub-optimal control; several examples are considered. The constrained optimal control depends on the initial state and new optimisation criteria must be put forward to deal with the case in which the initial state is unknown; the most common consist of minimising the cost that can result from the worst initial state. It is then shown how the controllers designed according to these criteria may be improved by using one's limited observation at time zero to place some constraints on the initial state. The Liapunov matrix equation plays an important part in calculating the cost of any control so reducing the computational effort in its solution is useful. It is shown how this can be done and it is of special relevance for distributed parameter systems with their states expressed as an infinite series of eigenfunctions; the results are applied to a diffusion equation example.
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\nFinally, it is shown how optimal control theory may be applied to the design of proportional-integral-derivative controllers. This is done from two standpoints and the resulting controllers are shown to be identical, though the second method of proof is valid for infinite dimensional systems. The results are then applied to a simple example and to a distributed population dynamics system. The practicality of the methods of the thesis are applied to a system with realistic parameters; recommendations are made as to the best approaches.
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Frank B. Dilley

ARE CONCLUSIVE PROOFS IRRELEVANT TO RELIGION? FOR CENTURIES the proofs for God's existence have been debated and their role in religious belie£-systems assessed. There have been those who claim that the proofs are conclusive, those who claim that they are invalid , and those who claim that they are valid but inconclusive . There have been disputes, furthermore, as to whether the logical base for religious belie£ systems should be sought in proofs and evidence rather than in "self-authenticating" religious experiences and acceptance o£ religious authority. But no one, until recently, ever contended that even if the proofs were conclusive it would be irrelevant to religious believers. People have thought that religion must be undergirded by proof, aan be undergirded by proof, and need not be undergirded by proof, but not that proof would be irrelevant. Now o£ course there are many senses in which a proof could be relevant to religious believers. The Vatican Council affirmed that i£ people denied that God could be known with certainty by the natural light of human reason they would be anathema. Thus it could have made quite serious practical differences to a believer whether he accepted the proofs or not. As philosophers we are not concerned with this sort of practical difference but with philosophical ones, and the question to be explored is: Is there something philosophically at stake for believers in this question o£ the proofs for the existence o£ God? I want to contend that Steven M. Cahn is utterly wrongheaded in his claim that religious believers have no real interest in philosophical proofs for the existence of God.1 I am assuming that this 1 Steven M. Cahn, " The Irrelevance to Religion of Philosophic Proofs for the Existence of God," The American Philosophical Quarterly (April, 1969), pp. 170-72. Further citations will be given by page number in the text. Cahn has 727 7~8 FRANK B. DILLEY is not a statistical claim to be settled by polls but rather a philosophical claim. Cahn makes no effort to produce questionnaire results from a proper sample of believers nor shall I. To examine this question, in the context of Cahn's contention , it is fortunately not necessary to resolve the question of the validity of the proofs. In an effort to dismiss them once for all, Cahn is willing to assume that the proofs are valid, for the sake of the argument. " Suppose we assume, contrary to what most philosophers, I among them, believe, that all of these proofs are valid. Let us grant the necessary existence (whatever that might mean) of the most perfect conceivable Being, a Being who is ali-good and is the designer and creator of the universe. What implications can be drawn from this fact which would be of relevance to human life? In other words, what difference would it make in men's lives if God existed?" (p. 170) In support of his surprising thesis that religious believers are right in ignoring the proofs, he gives two chief arguments, that religious believers make their decisions as to what beliefs to hold and what practices to follow on the basis of "self-validating " personal experiences which render the proofs unnecessary , and that the success of the proofs would not enable a newly converted unbeliever to know what beliefs and practices to choose. He therefore contends that the proofs are relevant for neither the believer nor the unbeliever. In examining his contention about the relevancy of proofs I propose to look at what religious belief systems say about this problem, an approach Cahn does not seem to care to use. In this connection it should be noted that his sources are curious, for he cites not one person who would be taken by anyone to be a normal religious believer. He cites Antony Flew, Wallace Matson, C. B. Martin, and Mordecai Kaplan, but not John reprinted this essay in his Philosophy of Religion (Harper & Row, 1970), and it can also be found in Philosophy in the Age of Crisis, ed. Eleanor Kuykendall (Harper & Row, 1970) and God, Man, and Rdigion: Readings in the Philosophy of Rdigion, ed. by Keith Yandell (McGraw-Hill, 1973). ARE CONCLUSIVE PROOFS IRRELEVANT...

Serge‐Christophe Kolm

Many policies use two categories of instruments: a financial incentive (most often a tax) for polluters, and direct undertakings or financing of some restorations or maintenances or improvements of environmental qualities. The financial consequences of such policies, when optimum, are important for considerations of public finance, decentralization (financial autonomy), and equity (must polluters pay?). They turn out essentially to depend upon the mathematical structure of, first, the environment function, i.e., the way in which qualities depend upon both deteriorating and improving activities, and, second, the various constraints of the problem. Constraints which can be expressed by functions homogeneous of any degree are shown to have no direct financial effect. Apart from the constraints' effects, budgetary equilibrium, surplus or deficit are respectively given by functions which present constant, decreasing, or increasing qualitative returns to scale, i.e., weighted homogeneity of degree zero, positive or negative. The opposite polar cases of cleaning and dilution types of improvement technology are presented, with some other mixed simple cases and a few examples of application of the results.

Martin Feldstein

A series of recent judicial decisions has focused public attention on the issue of local choice in the provision of public education. In Serrano vs. Priest, Rodriguez vs. San Antonio (1971), and similar cases in other states, the lower courts confirmed that education is a responsibility of the state government and held that local expenditures on education may not be a function of the taxable wealth of the local community.1 Although the United States Supreme Court has overturned these decisions in the appeal of Rodriguez vs. San Antonio (1973), the pressure to change the current system remains strong. The Supreme Court majority indicated that its decision reflected the limits of the federal constitutional authority and was not an approval of the status quo in educational finance. Litigation is now likely to shift to challenging the current methods as unconstitutional under state constitutions which, unlike the federal constitution, do deal specifically with education.2 Moreover, fundamental changes in the financing of local education may not require further pressure from the courts; state legislatures may seek to neutralize the effects of local wealth differences even if the current systems are not held to be unconstitutional. These judicial decisions and the ensuing legislative proposals run counter to the general economic view of local government finance. The basic presumption of economic analysis is that, because local governments can select different levels of service and because individuals can choose their area of residence, decentralized finance by local governments allows the provision of public services to reflect the variety of individual preferences for public services.3 Although the level of local spending may be nonoptimal because of intercommunity externalities and because of the method of local budget determination, fiscal decentralization still remains the only alternative to the insuperable problem of determining the optimal level of expenditure on a public service provided by a central government. In effect, autonomous decentralized financing of education provides a quasi market in which households can exercise their diverse preferences by their location decisions. This paper considers the problem of * Professor of economics, Harvard University. I am grateful to Charles Clotfelter for assistance with the statistical analysis, to Stephen Weiss for providing unpublished data on school expenditures, and to the Ford Foundation and National Science Foundation for financial support. I have benefited from discussions of an earlier version in seminars at Harvard, M.I.T., and Berkeley, and from comments by Noel Edelson, Eric Toder, and David Stern. An earlier and more complete discussion of this study was distributed as Harvard Institute of Economic Research paper no. 293, May 1973 (revised July 1973). 1 In Serrano vs. Priest, the landmark case in this area, the plaintiff and the courts were very much influenced by the line of argument and suggested remedies develope(l in John Coons et al. For a further discussion of the legal precedents, see Arthur Wise. 2 Almost immediately after the United States Supreme Court decision in Rodriguez vs. San Antonio, the New Jersey Supreme Court held that the current system of local finance violated the New Jersey state constitution. See Wise for a summary of the provisions of other state constitutions. I Charles Tiebout presented a formal analysis of the full efficiency of local government provision of public services under quite special conditions. See Wallace Oates and James Buchanan and Charles Goetz for a further discussion of these issues.

Edsger W. Dijkstra

The synchronization task between loosely coupled cyclic sequential processes (as can be distinguished in, for instance, operating systems) can be viewed as keeping the relation “the system is in a legitimate state” invariant. As a result, each individual process step that could possibly cause violation of that relation has to be preceded by a test deciding whether the process in question is allowed to proceed or has to be delayed. The resulting design is readily—and quite systematically—implemented if the different processes can be granted mutually exclusive access to a common store in which “the current system state” is recorded.

Thomas J. Graves

IGR and the New Federalism are synonymous. Former President Nixon summarized the New Federalism as "A cooperative venture among governments at all levels ... in which power, funds, and authority are channeled increas ingly to those governments which are closest to the people." IGR is essentially an art and primarily an exercise in the be havioral field. The thread which binds government officials intergovernmentally is the financing and administration of federal grants-in-aid and other forms of federal financial assis tance. Concern in the executive branch for IGR developed in the last 20 years; thus its roots in American governance are not yet very deep. The views and support of public interest groups as well as the sympathetic attention of Congress are essential to the pursuit of IGR goals and objectives in the executive branch. In relations between levels of government, partnership rather than paternalism must prevail. The rise of stronger chief executives and decentralization of power from Washington are noted. Confidence of Americans in their own government is essential to achieving a federal system working in all parts, well-managed and equitably financed.

David Harvey

Before geographers commit themselves to public policy, they need to pose two questions: what kind of geography and what kind of public policy. The evolution of the discipline, in terms both of its aims and its professional organization, must be seen as an adaptation to external conditions, particularly to the development of the corporate state with its emphasis on the 'national interest'. The corporate state forces education to be seen purely as investment in manpower and academic research becomes subservient to the state and is used to preserve and strengthen the status quo. There is here a potential conflict with the academic's sense of moral obligation, but in practice the conflict is resolved by the parochialism and elitism of the humanistic tradition. To help to move away from the corporate state and towards the 'incorporated state' in which men can control the social conditions of their own existence, geographers need to address their efforts towards understanding the tension between the humanistic tradition and the pervasive needs of the corporate state and thereby to learn how to exploit the contradictions within the corporate state itself. CAN geographers contribute successfully, meaningfully and effectively to the formation of public policy ? General Pinochet is a geographer by training, and by all accounts he is successfully putting geography into public policy. As President of the military Junta that overthrew the elected government of Salvador Allende in Chile on I September I973, General Pinochet does not approve of 'subversive' academic disciplines such as sociology, politics and even philosophy. He has asked that 'lessons in patriotism' be taught in all Chilean schools and universities and he is known to look with great favour upon the teaching of geography-such a subject is, he says, ideally suited to instruct the Chilean people in the virtues of patriotism and to convey to the people a sense of their true historic destiny. Since the military have taken full command of the universities and frequently supervise instruction in the schools, it appears that geography will become a very significant discipline in the Chilean educational system. General Pinochet is also actively changing the human geography of Chile. An example is here in order. The health care system of Chile has, for some time, comprised three distinct components: the rich paid for services on a 'free-market' basis; the middle classes made use of hospital-based medicine financed by private insurance schemes; while the lower classes and poor (some 60 per cent of the population) received free medical care in community-based health centres paid for out of a National Health Service.1 Under Allende, resources were switched from the first two sectors into the community health services which had previously been poorly financed and largely ignored. The geography of the health care system began to be transformed from a centralized, provider-controlled, hospital-centred system catering exclusively to the middle and upper classes, to a decentralized, community-controlled, free health care system primarily catering to the needs of the lower classes and the poor. This transformation did not occur without resistance-the providers of hospital-based medicine organized strikes to preserve the old social geography of health care against the emergence of the new. But during the Allende years the community health centres grew and flourished. Also, community control through the creation of Community Health Councils had a profound political impact and many aspects of life began to be organized around the community health centres. The emphasis also shifted from curative medicine (with all of its glamour and expensive paraphernalia) to preventive medicine which sought to treat medical care as something integral to a wide range of environmental issues (water supply, sewage disposal, and the like). The human geography of social contact, political power and distribution changed as hitherto never before, as the lower classes and poor people began to realize the potential for controlling the social conditions of their own existence.

L. Vaughn Blankenship

No abstract is available for this record.

Glenn M. Torrie, John P. Valleau

No abstract is available for this record.

Richard H. Rand

No abstract is available for this record.

A. G. Evans, S. M. Wiederhorn

No abstract is available for this record.

Burton Gummer

The American social welfare field is best characterized as a highly decentralized sphere of activity in which autonomous organizations define and pursue their goals in a fairly independent fashion. The complex nature of modern social problems, however, requires concerted action by a variety of organizations if effective solutions are to be developed. This conflict between the structural nature of the welfare field and the demands of the problems to be addressed has meant that social welfare planners have had to be concerned with the conditions affecting the willingness of independent organizations to engage in cooperative activities with each other. The purposes of the present paper are twofold: (1) To identify some of the major variables that affect the interorganizational activities of social welfare organizations; and (2) to describe the actual interorganizational patterns of one such organization, a county board of public assistance.

P. Hartung

No abstract is available for this record.

Petrus Hendrikus Maria Ruys

I. Economics.- 1. Introduction.- 1.1. Economic theory and economic environment.- 1.2. Private and public goods.- 1.3. The duality approach.- 1.4. Decentralized allocation mechanisms.- 2. Equilibrium in a system of economic relations.- 2.1. General equilibrium theory.- 2.2. Optimal allocations.- 2.3. An exchange economy.- 3. Production.- 3.1. Production sets and production multifunctions.- 3.2. Conditions in the production model.- 3.3. The price structure corresponding to a technology.- 3.4. Conditions in the price space.- 3.5. Preference orderings of inputs.- 3.6. Satiation for and dispensability of inputs.- 3.7. The demand- and the price-multifunction.- 4. Consumption and production of public goods.- 4.1. The consumption model.- 4.2. Private goods, public goods and externalities.- 4.3. Private goods, public goods and transaction costs.- 4.4. Social consumption and production.- 4.5. An economy with local public goods.- 5. Equilibrium in economies with private and public goods.- 5.1. The valuation representation of an economy.- 5.2. Equilibrium in an economy with public goods only.- 5.3. Equilibrium in an economy with private and public goods.- 5.4. The two-level price equilibrium.- 5.5. Financing the public sector.- 6. The organization of economic decisions.- 6.1. Allocation mechanisms.- 6.2. Centralization and decentralization.- 6.3. Procedures with a social preference ordering.- 6.4. Procedures with individual preference orderings.- 6.5. Multi-level organization.- 7. Extensions.- 7.1. Between values and resources.- 7.2. Theory of motion.- II. Mathematics.- 8. Basic mathematical notions and notations.- 8.1. Sets, relations and multifunctions.- 8.2. Continuity of multifunctions.- 8.3. Sets and algebraic operations in Rn.- 9. Sets and duality.- 9.1. Supporting hyperplanes and separating hyperplanes.- 9.2. Polar sets.- 9.3. Reflexive sets.- 9.4. Separation and intersection of sets.- 10. Multifunction and duality.- 10.1. Operations on multifunctions.- 10.2. Properties of multifunctions.- 10.3. Convex processes.- 10.4. Convex cone-interior processes.- 10.5. Convex star and convex aureole processes.- Summary.- References.

William J. Ledger

No abstract is available for this record.

T.D. Padrick, M.A. Gusinow

No abstract is available for this record.

J.L. Ferreira Mendes

1. Like many other countries, Portugal is facing numerous major problems related to regional unbalance and consequently to the location of economic activity. An unfortunate tendency has long been to judge an economy almost solely from the viewpoint of national production and consumption aggregats, without reference to the geographic division of economic activity.However the questions of social justice in the distributions of the fruits of economic development are as important and as different in terms of regions as in terms of social classes.Thus, at the present, increasingly attention is not being limited to the overall results of national development; the results are being accepted as satisfactory only if they concern the whole of the country, if each region is able to contribute to and participate in the national growth..On the other hand, Government activity to guide economic and social development involves not only the activity of policies and programmes but also the losely related process of formulation of goals and means and of appraisal of the results.If planning for regional development is intended to intervene realistically upon reality, it should be a continuous process in which the various units and levels of government subject the entire process to continuous review and evaluation leading to adjustment of plans, programmes and projects whenever necessary. Review and evaluation then form the transition to a new cycle of planning and decision making, moving ahead in time on the basis of a continuous stream of feedback information. At the same time, the fact is stressed that planning is actually a combination of plan formulation and plan inplementation.Institutional framework, in the modern sense of the process of achieving intended results through organizations, is a major factor at all levels - national, regional, sectoral and local. There was a time when proposals for new development pro. jects - particularly large programmes for resource development, new industries, improved education and health services - were considered only in terms of economic and technical feasibility.. After many unfortunate failures, institutional feasibility has come to be recognized as also an important dimension of planning.For evaluating the evolution of the objectives and means which have characterized Portuguese regional planning I rely in large measure on a general policy model formulated in the light of relevant theoretical and empirical considerations from economics and related social science disciplineOne of the basic attributes of this model is its distinction among three types of analytic regions: congested, potential, and backward. The advantaSe of these distinctions over the common division between "developed" and "underdeveloped" regions is that they come to grips directly with the problem of overconcentration of population and economic activity in some areas, a problem too often neglected in favour of studying the difficulties experienced by relatively underdeveloped regions.The circumstances which have given rise to the evolution of' Portuguese regional policy are compared with the assumptions of this basic model. Attention is given to the basic theoretical notions that have animated Portuguese thought concerning regional development and urban-rural integration, as well as to concrete measures which have been undertaken or which are envisaged for the future.On the basis of these considerations an effort is made to formulate a number of generalizations regarding the potential strenghts and difficulties of regional planning policy-making and a number of operationally feasible proposale are set forth for dealing with the difficulties.2. Portuguese regional planning was initiated on the basis of a deliberated attempt to deal rationally with spatial resource allocation, namely through the policy of "ordenamento do território".In the terminology of the Third Plan, the Portuguese approach maintains that the policy of "ordenamento do território" must find a practical compromise between regions depending on a policy of publicly induced growth and regions depending on a policy of induced public investment. On the one hand, it must give every opportunity, under conditions of lively competition, to strong regions whose potential benefits the whole of the country. On the other hand, it must seek to involve the weak regions in a process of development at first induced, then autonomous, in a manner which will enable them to participate in the current of modernization and expansion which characterizes our time.I try to show that the difficulties of Portuguese's backward regions are in large measure a result of a relative lack of benefits deriving from social investment. This is not to deny that economic investment in backward regions will produce advantages, but the effectiveness of such projects also depends on the degree to which the regions' human resources have been developed by social investment.Therefore, I stress that while there is official recognition of the needs to provide facilities for training surplus agricultural labor for employment in industry, there has not been adequate emphasis on the general problem of developing the human resources of these areas, despite the multitude of evidence concerning pronounced needs in this regard..On the other hand, a great deal has been said and written about moving industry and other economic activities to people, but policy-makers generally are more reluctant to urge the movement of people to job sources. Until now official Portuguese policy has been unable to oppose and to diminish the growth of a surprising number of rural individually-owned farms in the face of rapid technological change. This attitude, combined with inadequate social investment, has served to perpetuate the social and economic structures characteristic of backward agricultural regions of the center and the north including a surplus agricultural labor force.One of the principal arguments in favour of moving industry to backward regions, even at considerable expense in theform of subsidies or similar means, has been that the social costs involved in this type of action are less than the social costswhich would entail the uprooting of persons seeking employment in other regions as well as the increased congestion which would result in industrial agglomerations, However, the latter difficulty is now a necessary one, since migration can be channeled to intermediate or potential, rather than congested regions.On the other hand, the issue of uprooting residents of backward areas is a genuine problem since there is abundant evidence that the number of persons preferring to live in these regions is high in both absolute and relative terms.. Moreover, migration is not feasible for many persons because deficiencies in social investment in backward regions have limited the development of human resources and thus the possibility for their employment in other regions. Nevertheless, these arguments should not be used to discourage migration, since at the margin there are always persons ready and willing to migrate from backward regions.In general, it may be said that Portuguese regional policy has been facilitated by its distinction among three types of region, and that its over-all division of effort among the regions so as to induce growth in backward regions while limiting the growth of the Lisbon region and allowing for the expansion of potential regions has been substantially correct. Investment policywithin regions, however, must place relatively greater stress on social investment for backward regions. As to population policy, the new emphasis on encouraging interregional labor migration is a positive step away from the more conservative attitudes which prevailed heretofore.3. In any event, it is obvious that the value of the Portuguese regional development policy is related to the actual context in which it is to be applied. In this respect, the creation of regional organisms to participate in the formulation of regional priorities and in the regionalization of the government budget are of particular value in the elaboration of political ends and means.Unfortunately, earlier efforts in this direction were characterized by important defects. The prograps for regional action which were drawn up as planning guidelines for the various planning regions generally have been plans in name only. They have been for the most part inventories of regional conditions at a giventime. In addition, they have been deficient intipulating orders of priority and modes of finance for suggested future projects.Nevertheless, these programs have served to confront the Regional Planning Commissions with the need for horizontal consultation and coordination in regional terms as a complement to vertical planning by sectors; they have marked an initial, if not always successful, attempt to encourage cooperation among ministries and departments on common problems..Thus, in order to satisfy the objectives of regional development foregoing discussed, the need for reform of the structures - or at least changes in the institutional apparatus - become increasingly necessary. To avoid "sprinkling" of public funds it is not enough just to introduce new methods into the organizational bodies concerned; it becomes clear that something has to be done about the decision-making structures themselves. Two complementary requirements become apparent among others:- firstly, greater decentralization of decisions within the institutional apparatus and greater participation of people.- secondly, horizontal co-ordination in order to break up the vertical compartmentalizations and to create in that way the conditions for a better synthesis capable of inspiring better decisions.Just now, a new development Plan for Portugal is in preparation covering the period 1974-1979, Considering the methods of planning which have been employed in the country, one could

Stanley J. Kahrl

REVIEWS Alan H. Nelson. The Medieval English Stage: Corpus Christi Pageants and Plays. Chicago: University of Chicago Press, 1974. Pp. xiv + 274. $12.50. In his introductory chapter Alan Nelson states the thesis of his book as follows: “ I will argue from documentary evidence—much of it previ­ ously unknown—that the fifteenth-century English dramatic cycles de­ veloped out of fourteenth-century festival processions. A corollary of this argument is that the doctrinal relationship of the Corpus Christi dramatic play to the feast of Corpus Christi is almost entirely incidental” (p. 9). This thesis has hardly the revolutionary quality claimed for it by David Bevington on the dust jacket, particularly when the book in fact never discusses the plays themselves or their doctrinal content at all. Scholars at work in the field of medieval English drama have been saying for at least fifteen years, certainly since the appearance of O. B. Hardi­ son’s Christian Rite and Christian Drama in 1965, that the vernacular civic drama of England in the later Middle Ages represents a new begin­ ning, and is not to be understood in its formal aspects by describing it in terms of Latin liturgical plays. The efficient cause of such plays, to use Nelson’s own Aristotelian language, was certainly the context provided by the annual procession of Corpus Christi day. The value of this book lies in its continual insistence on the reality of that Corpus Christi procession, an event that has, in fact, been too often treated lightly. The records that Nelson cites from cities all over England remind those in protestant countries how impressive such a procession was, how involved were the preparations for its occurrence, and how much time and space it took. In pursuit of this interest Nelson traveled this past spring to Spain. His slides of one such procession shown at the conference on Medieval drama held at Leeds in September 1974 were a useful reminder as well of how much of an event the procession alone can be. Those already familiar with Nelson’s article on “Principles of Pro­ cessional Staging: York Cycle,” which appeared in Modern Philology in 1970 and was read at the Medieval Drama Seminar of the MLA meeting in 1969, which reappears here as Chapter II, will realize that more is at stake than simply a reinforcement of the connection between the Corpus Christi procession and the civic plays called by that name, whether pro­ duced then or on another occasion. In that article Nelson applied pure reason, using mathematical formulae, to the problem of what went on in York, and arrived at the conclusion that it was mathematically impossi386 Reviews 387 ble for the surviving York plays to have been put on in one day, let alone in a day which also contained a time-consuming Corpus Christi proces­ sion. It is likely that Nelson chose York as his point of attack because it has for so long dominated histories of medieval English drama as the paradigm of what happened throughout England. The immediate result of that article was to renew interest in the York records themselves. In “Two Studies of The York Corpus Christi Play” (Leeds Studies in English, 6 [1972], 63-111), Margaret Dorrell offered a rebuttal to Nelson’s argu­ ment, offering what to many has been convincing proof that, whether or not the plays could not have been done in one day at multiple stations according to the principles of logic, the fact is that that is exactly how they were put on. Nelson was thus himself forced back to the records if he was to sup­ port his attack on the traditional history of events in York. Thus his in­ sistence on his use of “documentary evidence—much of it previously un­ known.” Certainly he is to be commended for the labor involved in con­ sulting a wide range of documents, from an impressive number of towns and cities in England. Not only has he consulted the local records of such obvious dramatic centers as York, Beverley, Lincoln, Norwich, Coventry, Chester, and London, but in Chapter 11 he has amassed a con­ siderable body of material not easily obtainable before...

Elliott H. Lieb, Mary Beth Ruskai

We prove several theorems about quantum-mechanical entropy, in particular, that it is strongly subadditive.

É. Sanchez-Palencia

No abstract is available for this record.

Alan P. Lightman, David L. Lee

No abstract is available for this record.

N. P. Red’kin

No abstract is available for this record.

Sidney Shindell

No abstract is available for this record.