BOOK REVIEWS 345

highlight ofthe chapter is the proofofthe exis-tence of at least two geometrically distinct pe-riodic solutions on each energy surface; gener-ically it is shown that there are infinitely many,except in the case of one degree of freedom.The chapter ends with a list of open problems.

This book is a very valuable addition to theliterature on Hamiltonian systems. It presentsin a well written and coherent form materialthat is scattered in a lot of papers and famil-iarizes the reader with one of the great successstories in modern Hamiltonian dynamics. Theauthor has gone to great pains to make thematerial accessible to a wide audience by in-cluding almost all necessary background. Thebook succeeds as both a reference work and asource of information for an advanced course.It should be on the bookshelfof all those inter-ested in mechanics and symplectic geometry.

TUDOR RATIUUniversi of California at Santa Cruz

Engineering Risks: Evaluation and Valua-tion. By Ulrich Hauptmanns and WolfgangWemer. Springer-Verlag, Berlin, 1991. xii +246 pp. $89.00. ISBN 0-387-52195-X.

The book Engineering, Risks: Evaluationand Valuation by Hauptmanns and Werner isan English translation and a thoroughly re-vised edition of the German original that ap-peared in 1987.

This monograph considers the risks relatedto complex industrial plants. Special attentionis paid to energy production and the chem-ical industry. As the subtitle indicates, thebook evaluates other risk studies in additionto giving valuations of risks related to dif-ferent energy production technologies. Thebook also gives a rather comprehensive reviewon recent safety studies performed in nuclearpower plants in the United States and Ger-many.

In the introduction the authors define thefundamental concepts needed in the rest ofthe book. In mathematical terms there is oneparticularly interesting definition. Risk is de-fined as the triplet

Ri (si, hi, Xi)

where s is the event sequence leading todamage, h is the corresponding expected fre-

quency of occurrence, and X represents theconsequences. The identification of the eventsequence reflects the pragmatic approach ofthe book. It is vital in risk engineering to iden-tify the sequences that lead to damage in or-der to be able to prevent accidents and to takerestrictive measures after an accident. Simi-larly, it is necessary to know the sequence inorder to assess the probabilities. In the ab-sence of sufficient statistics it is not possibleto calculate or estimate the probabilities basedon historical data only. It is necessary to knowthe physical and chemical processes effectivein the event sequences.

In the second chapter different methodsof risk analysis are presented. Many aspectsof the processes are difficult to model, andhence, subjective expert opinions must beused. One of the most difficult factors to es-timate is the human element in the process.Some methods are described for the analysisand prediction of human error rates.The descriptions in this chapter are rather

superficial from the mathematical point ofview. The presentation serves more practi-cal purposes. The text is directed to users ofrisk analyses who are usually nonmathemati-cians. In order to acquaint oneself more thor-oughly with the mathematics or the methods,the reader is referred to the references men-tioned at the end of the chapter.The third chapter describes the risk anal-

yses that have been performed in nuclear in-stallations. The emphasis is on the conse-quences on employees and population of ac-cidents in nuclear power plants. These analy-ses are extremely important and difficult. Onehas to handle very complicated processes andvery rare incidents with severe consequences.The (fortunately) scarce data on accidents indifferent nuclear installations and short run-ning history of these installations togetherwith very small probabilities (mainly of the or-der 10-6 or 10-7) renders the estimated prob-abilities very uncertain with wide confidenceintervals. The assessing of these probabilitiesand their confidence intervals requires deepknowledge of the processes that take place innuclear power plants. Methods from math-ematics, statistics, and reliability theory canonly play an assisting role. The task, however,is so important that all the help that mathe-matics and statistics can give in estimating fre-quencies of rare events is welcomed.

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346 BOOK REVIEWS

The fourth chapter concentrates on riskstudies for process plants. There are two spe-cific studies that are evaluated. The one per-formed in the U.K. examines the risks relatedto a whole industrial complex. The other (per-formed in the Netherlands) deals with differ-ent chemical plants. Impacts from other in-stallations were not included. As a conclusionit is worth mentioning that the risk studies inchemical installations are far behind those ofnuclear installations. This reflects the worrythat is felt towards the unknown risks of nu-clear power compared with those more famil-iar of process plants. These risks, however,can be very fatal. We only have to remem-ber Seveso and Bhopal to convince ourselvesabout this. The authors give the following listof requirements to be satisfied in order thatrisk analysis for process plants reach the levelof the nuclear installations:

Systematic collection of reliability datafor components in process plants underadequate quality assurance.Systematic evaluation of operating expe-rience in order to obtain reliable infor-mation on accident-initiating events andtheir frequencies of occurrence.Development and validation of modelsfor physical and chemical processes likethe discharge of multi-phase and multi-component mixtures and for vapourcloud explosions.Development of models for the at-mospheric dispersion of explosible andtoxic substances, accounting for surfaceroughness and buildings in the vicinity ofthe location of release.Determination of dose-effect relation-ships for important toxic substances.

The fifth chapter deals with risk compar-isons for nuclear and conventional energyconversion systems. The authors again referto numerous earlier studies made on differentenergy production technologies. The studiesare not limited to production plants. Also theconstruction of the plants, mining of the rawmaterials as well as all transportation and han-dling ofwaste are treated. Methodologies likeinput-output analysis also attempt to take intoaccount the other segments in the economy inorder to calculate the comprehensive effect ofeach technology to produce energy.The sixth chapter considers the prevail-

ing practices of appraisal of safety in nuclear

power plants. The probabilistic approach isvisible in the definition of different safety lev-els. The ideology behind this is that an eventsequence that leads to severe consequencesmust not have a high probability of occur-rence. This gives a natural scale for assess-ing the decreasing allowed probability with in-creasing severity of consequences.The book reveals a huge need for further

research on methods to be used in risk and re-liability analysis in both nuclear and processplants. Systematic gathering and analysis ofdata on events that do not lead to a catas-trophe are also needed. The book gives us arather comprehensive picture of the presentstate of art and of the deficiencies therein.The importance of strict scientific approach isall the more important when we take into ac-count the fact that these results are used in po-litical decision-making when future guidelinesfor energy production are being made. Thisis not a book on applied mathematics, but itdeals with one field where the need for appliedmathematics is apparent.

MATH RUOHONENThe Sampo GroupTurku, Finland

Nonlinear Elliptic and Evolution Problemsand Their Finite Element Approximations.By A. enigek. Translation editor J. R. White-man. Academic Press Ltd., UK, 1990. xix +422 pp. $94.50. ISBN 0-12-779560-X. Com-putational Mathematics and Applications.

As a summary of frustrating experiences,the First Law of Applied Mathematics an-nounces: "Nothing Fits." Thus, if you wantto quote a result about convergence of finiteelement approximations, you are likely to findit for a basic polygonal domain, whereas yourdomain has a curved boundary; or, having fi-nally found a paper dealing with your typeof nonlinearity, the initial data are there as-sumed smoother than allowed in your appli-cation, and so on.

Attempting partially to defy the First Law,the present treatise by Professor eniek isa synthesis of many investigations devotedto isolating and conquering specific difficul-ties. Let me briefly describe a culminat-ing Super Theorem of the book in the time-dependent situation. (There is a correspond-

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