The Geometry of Population Genetics by E. AkinReview by: Philip HolgateBiometrics, Vol. 38, No. 1 (Mar., 1982), pp. 286-287Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2530321 .

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286 286 286 Biometrics, March 1982 Biometrics, March 1982 Biometrics, March 1982

also refer to a further discussion by Dietz in the 1981 publication mentioned earlier in this review.

Some of the papers are directed to more general queries on the 'right' approach to mathematical mod- els in biology. Ricciardi examines a bone of contention in the literature, namely the rival claims of the Ito and Stratonovich calculi when stochastic noise is added to nonlinear deterministic equations used as models of population processes; but any ambiguities are of course the consequence of imprecise specification or approximation of the stochastic model being used.

One could go on discussing all the various contribu- tions, but it may suffice to end by referring to the final biographical sketch by Borsellino of Volterra, who was born in 1860 and became Professor of Rational Mechanics at Pisa when 23. He joined the University of Rome in 1900, and gave a talk 'on the attempts to apply mathematics to biological and social sciences', some 25 years before his classical paper on mathemat- ical population biology published by the Accademia dei Lincei when he was its President.

M. S. BARTLETr Priory Orchard, Priory Avenue,

Totnes, Devon, England.

ASHTON, E. H. and HOLMES, R. L. (eds) Perspec- tives xn Primate Biology. Symposia of the Zoological Society of London, No. 46. Academic Press, London, 1981, 424 pp. £28.40/$68.50.

This volume records the 75th birthday celebrations for Lord Zuckerman, formerly Chief Scientific Advisor to the British Government. Most of the 14 papers pre- sented at the symposium are of exceptionally high quality, and in mentioning a selection of particular biometrical interest I should not fail to note that a few papers, such as the one on insect hormones and neurosecretion, may pass unnoticed in a book with a title concerned with primate biology. This diversity reflects part of the vast range of Lord Zuckerman's activities.

For a biometrician, the core of the book will lie in the two sections on Primate Growth and on Primate Evolution, Biometrical Morphology, and Palaeontol- ogy. In the first section Widdowson compares growth in animals of very different sizes with special emphasis on growth before birth. Healy and Tanner present a condensed account of the classical problem of size and shape in relation to growth and form, with the com- ment that only through applying multivariate techni- ques to real data can their strengths and weaknesses can be discovered. Moore analyses facial growth in primates, comparing apes and men. In the second section there is a long (59 pages) paper by Ashton on the Australopithecinae, demonstrating that they are not intermediate between apes and men but share characters with both; multivariate methods play a substantive role in this demonstration. Oxnard and Yang go beyond such methods to look at the theories of stress-bearing in cancellous bone as reflected in the trabecular arrangements within the bone. Using

also refer to a further discussion by Dietz in the 1981 publication mentioned earlier in this review.

Some of the papers are directed to more general queries on the 'right' approach to mathematical mod- els in biology. Ricciardi examines a bone of contention in the literature, namely the rival claims of the Ito and Stratonovich calculi when stochastic noise is added to nonlinear deterministic equations used as models of population processes; but any ambiguities are of course the consequence of imprecise specification or approximation of the stochastic model being used.

One could go on discussing all the various contribu- tions, but it may suffice to end by referring to the final biographical sketch by Borsellino of Volterra, who was born in 1860 and became Professor of Rational Mechanics at Pisa when 23. He joined the University of Rome in 1900, and gave a talk 'on the attempts to apply mathematics to biological and social sciences', some 25 years before his classical paper on mathemat- ical population biology published by the Accademia dei Lincei when he was its President.

M. S. BARTLETr Priory Orchard, Priory Avenue,

Totnes, Devon, England.

ASHTON, E. H. and HOLMES, R. L. (eds) Perspec- tives xn Primate Biology. Symposia of the Zoological Society of London, No. 46. Academic Press, London, 1981, 424 pp. £28.40/$68.50.

This volume records the 75th birthday celebrations for Lord Zuckerman, formerly Chief Scientific Advisor to the British Government. Most of the 14 papers pre- sented at the symposium are of exceptionally high quality, and in mentioning a selection of particular biometrical interest I should not fail to note that a few papers, such as the one on insect hormones and neurosecretion, may pass unnoticed in a book with a title concerned with primate biology. This diversity reflects part of the vast range of Lord Zuckerman's activities.

For a biometrician, the core of the book will lie in the two sections on Primate Growth and on Primate Evolution, Biometrical Morphology, and Palaeontol- ogy. In the first section Widdowson compares growth in animals of very different sizes with special emphasis on growth before birth. Healy and Tanner present a condensed account of the classical problem of size and shape in relation to growth and form, with the com- ment that only through applying multivariate techni- ques to real data can their strengths and weaknesses can be discovered. Moore analyses facial growth in primates, comparing apes and men. In the second section there is a long (59 pages) paper by Ashton on the Australopithecinae, demonstrating that they are not intermediate between apes and men but share characters with both; multivariate methods play a substantive role in this demonstration. Oxnard and Yang go beyond such methods to look at the theories of stress-bearing in cancellous bone as reflected in the trabecular arrangements within the bone. Using

also refer to a further discussion by Dietz in the 1981 publication mentioned earlier in this review.

Some of the papers are directed to more general queries on the 'right' approach to mathematical mod- els in biology. Ricciardi examines a bone of contention in the literature, namely the rival claims of the Ito and Stratonovich calculi when stochastic noise is added to nonlinear deterministic equations used as models of population processes; but any ambiguities are of course the consequence of imprecise specification or approximation of the stochastic model being used.

One could go on discussing all the various contribu- tions, but it may suffice to end by referring to the final biographical sketch by Borsellino of Volterra, who was born in 1860 and became Professor of Rational Mechanics at Pisa when 23. He joined the University of Rome in 1900, and gave a talk 'on the attempts to apply mathematics to biological and social sciences', some 25 years before his classical paper on mathemat- ical population biology published by the Accademia dei Lincei when he was its President.

M. S. BARTLETr Priory Orchard, Priory Avenue,

Totnes, Devon, England.

ASHTON, E. H. and HOLMES, R. L. (eds) Perspec- tives xn Primate Biology. Symposia of the Zoological Society of London, No. 46. Academic Press, London, 1981, 424 pp. £28.40/$68.50.

This volume records the 75th birthday celebrations for Lord Zuckerman, formerly Chief Scientific Advisor to the British Government. Most of the 14 papers pre- sented at the symposium are of exceptionally high quality, and in mentioning a selection of particular biometrical interest I should not fail to note that a few papers, such as the one on insect hormones and neurosecretion, may pass unnoticed in a book with a title concerned with primate biology. This diversity reflects part of the vast range of Lord Zuckerman's activities.

For a biometrician, the core of the book will lie in the two sections on Primate Growth and on Primate Evolution, Biometrical Morphology, and Palaeontol- ogy. In the first section Widdowson compares growth in animals of very different sizes with special emphasis on growth before birth. Healy and Tanner present a condensed account of the classical problem of size and shape in relation to growth and form, with the com- ment that only through applying multivariate techni- ques to real data can their strengths and weaknesses can be discovered. Moore analyses facial growth in primates, comparing apes and men. In the second section there is a long (59 pages) paper by Ashton on the Australopithecinae, demonstrating that they are not intermediate between apes and men but share characters with both; multivariate methods play a substantive role in this demonstration. Oxnard and Yang go beyond such methods to look at the theories of stress-bearing in cancellous bone as reflected in the trabecular arrangements within the bone. Using

Fourier transformations they have related the often orthogonal patterns in trabecular architecture to the postures and activities of the primates.

Lewis considers the functional morphology of the evolving foot and notes that the assignment of Olduvai hominid 8 to Homo habilis seems to be quite indefen- sible. The paper by Joysey on molecular evolution and vertebrate phylogeny, calling attention to the unrelia- bility of the molecular clock, echoes a view that, at least by implication, emerges in many accounts of the clash between cladists and pheneticists. This view picks out the organisms for which molecular methods are least successful in securing a phylogenetic placing as precisely those most difflcult to place by cladistic or phenetic methods. Joysey draws attention to the prob- lems posed by convergent and parallel evolution, problems which, though old, still cause trouble for taxonomists. He notes that the clustering technique which gave the 'best' zoological picture placed the horse among the primates, although the cetaceans, the apes, and the artiodactyls formed stable clusters what- ever clustering techniques were employed. Evidently, the cladist-pheneticist wrangle is unlikely to be resol- ved either soon or quietly.

The symposium volume is attractively produced and illustrated with photographs of Lord Zuckerman, Pro- fessor Harrison, and 30 other contributors and con- celebrants.

I find one major omission in this volume disturbing, though it is touched on in Harrison's closing address. Possibly the most important and fruitful work that Lord Zuckerman has done is to cast a critical eye on the pretensions of scientific advisors wielding such oversimplifications as 'the energy shortage' and 'the missile gap' to justify expensive and contentious solu- tions. The harm done by arrogant high-technology solutions to such problems is, I believe, casting a pall of discredit over the public's judgement of science for which a heavy price will in the long run be paid, and in Britain, with a string of such failures to its discredit, the price is already being paid in the diminished willingness of the public to fund a 'Science' that it increasingly sees as hostile when effective, and derisor- ily inept when ineffective. I cannot do better than recommend readers of the symposium volume to read, by way of supplement, Lord Zuckerman's own ac- count of these matters (Science Advisers, Scientific Advisers, and Nuclear Weapons, 1980, The Menard Press, London; and Proceedings of the American Philosophical Society, 1980,124; see also The Times, 21 January 1980 for a condensed version of the essay).

R. E. BLAcKrrH Zoology Department,

Trinity College, Dublin-2, Ireland.

AKIN, E. The Geometry of Population Genetics. Lecture Notes in Biomathematics, Vol. 31. Springer- Verlag, Berlin, 1980, 205 pp. $14.00.

The genetic composition of a population can be rep- resented by a point in a multidimensional space, in

Fourier transformations they have related the often orthogonal patterns in trabecular architecture to the postures and activities of the primates.

Lewis considers the functional morphology of the evolving foot and notes that the assignment of Olduvai hominid 8 to Homo habilis seems to be quite indefen- sible. The paper by Joysey on molecular evolution and vertebrate phylogeny, calling attention to the unrelia- bility of the molecular clock, echoes a view that, at least by implication, emerges in many accounts of the clash between cladists and pheneticists. This view picks out the organisms for which molecular methods are least successful in securing a phylogenetic placing as precisely those most difflcult to place by cladistic or phenetic methods. Joysey draws attention to the prob- lems posed by convergent and parallel evolution, problems which, though old, still cause trouble for taxonomists. He notes that the clustering technique which gave the 'best' zoological picture placed the horse among the primates, although the cetaceans, the apes, and the artiodactyls formed stable clusters what- ever clustering techniques were employed. Evidently, the cladist-pheneticist wrangle is unlikely to be resol- ved either soon or quietly.

The symposium volume is attractively produced and illustrated with photographs of Lord Zuckerman, Pro- fessor Harrison, and 30 other contributors and con- celebrants.

I find one major omission in this volume disturbing, though it is touched on in Harrison's closing address. Possibly the most important and fruitful work that Lord Zuckerman has done is to cast a critical eye on the pretensions of scientific advisors wielding such oversimplifications as 'the energy shortage' and 'the missile gap' to justify expensive and contentious solu- tions. The harm done by arrogant high-technology solutions to such problems is, I believe, casting a pall of discredit over the public's judgement of science for which a heavy price will in the long run be paid, and in Britain, with a string of such failures to its discredit, the price is already being paid in the diminished willingness of the public to fund a 'Science' that it increasingly sees as hostile when effective, and derisor- ily inept when ineffective. I cannot do better than recommend readers of the symposium volume to read, by way of supplement, Lord Zuckerman's own ac- count of these matters (Science Advisers, Scientific Advisers, and Nuclear Weapons, 1980, The Menard Press, London; and Proceedings of the American Philosophical Society, 1980,124; see also The Times, 21 January 1980 for a condensed version of the essay).

R. E. BLAcKrrH Zoology Department,

Trinity College, Dublin-2, Ireland.

AKIN, E. The Geometry of Population Genetics. Lecture Notes in Biomathematics, Vol. 31. Springer- Verlag, Berlin, 1980, 205 pp. $14.00.

The genetic composition of a population can be rep- resented by a point in a multidimensional space, in

Fourier transformations they have related the often orthogonal patterns in trabecular architecture to the postures and activities of the primates.

Lewis considers the functional morphology of the evolving foot and notes that the assignment of Olduvai hominid 8 to Homo habilis seems to be quite indefen- sible. The paper by Joysey on molecular evolution and vertebrate phylogeny, calling attention to the unrelia- bility of the molecular clock, echoes a view that, at least by implication, emerges in many accounts of the clash between cladists and pheneticists. This view picks out the organisms for which molecular methods are least successful in securing a phylogenetic placing as precisely those most difflcult to place by cladistic or phenetic methods. Joysey draws attention to the prob- lems posed by convergent and parallel evolution, problems which, though old, still cause trouble for taxonomists. He notes that the clustering technique which gave the 'best' zoological picture placed the horse among the primates, although the cetaceans, the apes, and the artiodactyls formed stable clusters what- ever clustering techniques were employed. Evidently, the cladist-pheneticist wrangle is unlikely to be resol- ved either soon or quietly.

The symposium volume is attractively produced and illustrated with photographs of Lord Zuckerman, Pro- fessor Harrison, and 30 other contributors and con- celebrants.

I find one major omission in this volume disturbing, though it is touched on in Harrison's closing address. Possibly the most important and fruitful work that Lord Zuckerman has done is to cast a critical eye on the pretensions of scientific advisors wielding such oversimplifications as 'the energy shortage' and 'the missile gap' to justify expensive and contentious solu- tions. The harm done by arrogant high-technology solutions to such problems is, I believe, casting a pall of discredit over the public's judgement of science for which a heavy price will in the long run be paid, and in Britain, with a string of such failures to its discredit, the price is already being paid in the diminished willingness of the public to fund a 'Science' that it increasingly sees as hostile when effective, and derisor- ily inept when ineffective. I cannot do better than recommend readers of the symposium volume to read, by way of supplement, Lord Zuckerman's own ac- count of these matters (Science Advisers, Scientific Advisers, and Nuclear Weapons, 1980, The Menard Press, London; and Proceedings of the American Philosophical Society, 1980,124; see also The Times, 21 January 1980 for a condensed version of the essay).

R. E. BLAcKrrH Zoology Department,

Trinity College, Dublin-2, Ireland.

AKIN, E. The Geometry of Population Genetics. Lecture Notes in Biomathematics, Vol. 31. Springer- Verlag, Berlin, 1980, 205 pp. $14.00.

The genetic composition of a population can be rep- resented by a point in a multidimensional space, in

This content downloaded from 62.122.73.86 on Wed, 25 Jun 2014 03:58:45 AMAll use subject to JSTOR Terms and Conditions

Book Reviews Book Reviews Book Reviews Book Reviews 287 287 287 287

which the coordinates stand for the numbers, or prop- ortions, of each type. The way in which these numbers change continuously in time, under the influences of differential viability, the chromosome breakage and recombination, and mutation, is described by the curve traced out by the point moving under these influences.

Differential geometry is therefore the natural setting for the mathematical study of this class of problems. It is remarkable that we have had to wait so long for the development of a coherent theory of the differential geometry of population genetics. A start was made by S. Shashahani (1979, Memoirs of the American Mathematical Society, No. 211), and the present monograph continues it. It contains four chapters, of which I and III are addressed by the author to 'the biologist'. I doubt whether they will be understood by many readers without postgraduate mathematics. The stylized used of 'biologists' to describe anyone else is a hindrance to communication that should be discarded.

These two chapters do however constitute a distinct stratum of the book, in which the author reviews the relevant parts of differential geometry, from a modern point of view. They show how selection, recombina- tion and mutation are related to differential geometric concepts in general, and discuss the specializations that are required in the genetic context. One of the crucial ideas is the Shashahani metric, which is in fact a manifestation of the x2 measure of genetic distance, but which has the differential geometric property that a vector field (representing a set of forces tending to change the population) is the gradient of some func- tion f with respect to this metric if the relative rate of change of xi is proportional to at/axi, where xi is the frequency of Type i. There is a very interesting discus- sion of the rdle of entropy in population genetics, of the meaning and implications of linkage (dis)equilib- rium, and of the ideas surrounding the belief that in 'real' populations, there is some function that can meaningfully be called 'fitness', that must increase with time. Stability is also considered.

Chapters II and IV, which the author calls the 'heart of the work', are disappointing in the sense that the detailed geometrical analysis contained in them, while it elucidates some of the mathematical problems raised elsewhere in the work, does not present these results in the form of biologically interpretable conclusions.

Akin's work is a substantial achievement in the extent to which it has brought together differential geometry and population genetics. The author's mas- tery of his material in both fields has led him to a relaxed and discursive style (in Chapters I and III) which makes his work a pleasure to read. At the same time it conveys the tantalizing feeling that in introduc- ing us to the range of ideas contained in this monog- raph, Akin has brought us to the brink of very impor- tant advances in the future. In assisting others to join in their elaboration, The Geometn of Population Genetics could be one of the seminal works of its decade.

PHILIP HOLGATE Birkbeck College,

University of London, - London, England.

which the coordinates stand for the numbers, or prop- ortions, of each type. The way in which these numbers change continuously in time, under the influences of differential viability, the chromosome breakage and recombination, and mutation, is described by the curve traced out by the point moving under these influences.

Differential geometry is therefore the natural setting for the mathematical study of this class of problems. It is remarkable that we have had to wait so long for the development of a coherent theory of the differential geometry of population genetics. A start was made by S. Shashahani (1979, Memoirs of the American Mathematical Society, No. 211), and the present monograph continues it. It contains four chapters, of which I and III are addressed by the author to 'the biologist'. I doubt whether they will be understood by many readers without postgraduate mathematics. The stylized used of 'biologists' to describe anyone else is a hindrance to communication that should be discarded.

These two chapters do however constitute a distinct stratum of the book, in which the author reviews the relevant parts of differential geometry, from a modern point of view. They show how selection, recombina- tion and mutation are related to differential geometric concepts in general, and discuss the specializations that are required in the genetic context. One of the crucial ideas is the Shashahani metric, which is in fact a manifestation of the x2 measure of genetic distance, but which has the differential geometric property that a vector field (representing a set of forces tending to change the population) is the gradient of some func- tion f with respect to this metric if the relative rate of change of xi is proportional to at/axi, where xi is the frequency of Type i. There is a very interesting discus- sion of the rdle of entropy in population genetics, of the meaning and implications of linkage (dis)equilib- rium, and of the ideas surrounding the belief that in 'real' populations, there is some function that can meaningfully be called 'fitness', that must increase with time. Stability is also considered.

Chapters II and IV, which the author calls the 'heart of the work', are disappointing in the sense that the detailed geometrical analysis contained in them, while it elucidates some of the mathematical problems raised elsewhere in the work, does not present these results in the form of biologically interpretable conclusions.

Akin's work is a substantial achievement in the extent to which it has brought together differential geometry and population genetics. The author's mas- tery of his material in both fields has led him to a relaxed and discursive style (in Chapters I and III) which makes his work a pleasure to read. At the same time it conveys the tantalizing feeling that in introduc- ing us to the range of ideas contained in this monog- raph, Akin has brought us to the brink of very impor- tant advances in the future. In assisting others to join in their elaboration, The Geometn of Population Genetics could be one of the seminal works of its decade.

PHILIP HOLGATE Birkbeck College,

University of London, - London, England.

which the coordinates stand for the numbers, or prop- ortions, of each type. The way in which these numbers change continuously in time, under the influences of differential viability, the chromosome breakage and recombination, and mutation, is described by the curve traced out by the point moving under these influences.

Differential geometry is therefore the natural setting for the mathematical study of this class of problems. It is remarkable that we have had to wait so long for the development of a coherent theory of the differential geometry of population genetics. A start was made by S. Shashahani (1979, Memoirs of the American Mathematical Society, No. 211), and the present monograph continues it. It contains four chapters, of which I and III are addressed by the author to 'the biologist'. I doubt whether they will be understood by many readers without postgraduate mathematics. The stylized used of 'biologists' to describe anyone else is a hindrance to communication that should be discarded.

These two chapters do however constitute a distinct stratum of the book, in which the author reviews the relevant parts of differential geometry, from a modern point of view. They show how selection, recombina- tion and mutation are related to differential geometric concepts in general, and discuss the specializations that are required in the genetic context. One of the crucial ideas is the Shashahani metric, which is in fact a manifestation of the x2 measure of genetic distance, but which has the differential geometric property that a vector field (representing a set of forces tending to change the population) is the gradient of some func- tion f with respect to this metric if the relative rate of change of xi is proportional to at/axi, where xi is the frequency of Type i. There is a very interesting discus- sion of the rdle of entropy in population genetics, of the meaning and implications of linkage (dis)equilib- rium, and of the ideas surrounding the belief that in 'real' populations, there is some function that can meaningfully be called 'fitness', that must increase with time. Stability is also considered.

Chapters II and IV, which the author calls the 'heart of the work', are disappointing in the sense that the detailed geometrical analysis contained in them, while it elucidates some of the mathematical problems raised elsewhere in the work, does not present these results in the form of biologically interpretable conclusions.

Akin's work is a substantial achievement in the extent to which it has brought together differential geometry and population genetics. The author's mas- tery of his material in both fields has led him to a relaxed and discursive style (in Chapters I and III) which makes his work a pleasure to read. At the same time it conveys the tantalizing feeling that in introduc- ing us to the range of ideas contained in this monog- raph, Akin has brought us to the brink of very impor- tant advances in the future. In assisting others to join in their elaboration, The Geometn of Population Genetics could be one of the seminal works of its decade.

PHILIP HOLGATE Birkbeck College,

University of London, - London, England.

which the coordinates stand for the numbers, or prop- ortions, of each type. The way in which these numbers change continuously in time, under the influences of differential viability, the chromosome breakage and recombination, and mutation, is described by the curve traced out by the point moving under these influences.

Differential geometry is therefore the natural setting for the mathematical study of this class of problems. It is remarkable that we have had to wait so long for the development of a coherent theory of the differential geometry of population genetics. A start was made by S. Shashahani (1979, Memoirs of the American Mathematical Society, No. 211), and the present monograph continues it. It contains four chapters, of which I and III are addressed by the author to 'the biologist'. I doubt whether they will be understood by many readers without postgraduate mathematics. The stylized used of 'biologists' to describe anyone else is a hindrance to communication that should be discarded.

These two chapters do however constitute a distinct stratum of the book, in which the author reviews the relevant parts of differential geometry, from a modern point of view. They show how selection, recombina- tion and mutation are related to differential geometric concepts in general, and discuss the specializations that are required in the genetic context. One of the crucial ideas is the Shashahani metric, which is in fact a manifestation of the x2 measure of genetic distance, but which has the differential geometric property that a vector field (representing a set of forces tending to change the population) is the gradient of some func- tion f with respect to this metric if the relative rate of change of xi is proportional to at/axi, where xi is the frequency of Type i. There is a very interesting discus- sion of the rdle of entropy in population genetics, of the meaning and implications of linkage (dis)equilib- rium, and of the ideas surrounding the belief that in 'real' populations, there is some function that can meaningfully be called 'fitness', that must increase with time. Stability is also considered.

Chapters II and IV, which the author calls the 'heart of the work', are disappointing in the sense that the detailed geometrical analysis contained in them, while it elucidates some of the mathematical problems raised elsewhere in the work, does not present these results in the form of biologically interpretable conclusions.

Akin's work is a substantial achievement in the extent to which it has brought together differential geometry and population genetics. The author's mas- tery of his material in both fields has led him to a relaxed and discursive style (in Chapters I and III) which makes his work a pleasure to read. At the same time it conveys the tantalizing feeling that in introduc- ing us to the range of ideas contained in this monog- raph, Akin has brought us to the brink of very impor- tant advances in the future. In assisting others to join in their elaboration, The Geometn of Population Genetics could be one of the seminal works of its decade.

PHILIP HOLGATE Birkbeck College,

University of London, - London, England.

an der HEIDEN, UWE. Analysis of Neural Networks. Lecture Notes in Biomathematics, Vol. 35. Springer- Verlag, Berlin, 1980, 157 pp. £13.00.

In this monograph the author sets up a fairly general system of nonlinear integral equations to model the relationship of spike activity and action potential in a neuron. The model includes the inhibitory-excitatory effects of neurons on each other in systems with possibly very large numbers of neurons. The treat- ment is sufficiently general to encompass several spe- cial cases that have previously been studied.

Despite references to a number of concrete exam- ples, this book consists mainly of classical functional analysis. It is primarily concerned with obtaining condi- tions on the functions in the model which ensure the existence of stationary solutions, whether they are unique and whether they are stable. The treatment has no statistical or stochastic elements, so that the book has little of interest to statisticians as such.

A. G. HAWKES Department of Statistics,

University College of Swansea, Swansea, Wales.

MOHSENIN, N. N. Thermal Properties of Foods and Agricultural Materials. Gordon and Breach, New York, 1980, 407 pp. $92.00.

This is a comprehensive review of experiments and techniques to establish relafionships describing the heating and cooling of a wide range of materials, and their effects. The six chapters cover basic concepts and theory, determination of specific heat and other ther- mal properties, and application to practical problems, with many numerical examples. Regression, linear and nonlinear, is widely used, as are graphical methods, mainly for predicting the effects of conditions such as temperature and moisture. The use of nondimensional groups to reduce the number of predictor variates is a recurring feature. The Appendix, containing tables, could be a useful source of data for examples on regression and curve-fitting.

C. R. BAINES Long Ashton Research Station,

Bristol, England.

CLIFF, A. D. and ORD, J. K. Spatial Processes, Models and Applications. Methuen, Andover, Hamp- shire, England, 1981, 266 pp. £12.50.

Interest in the analysis of spatial processes has grown rapidly, with studies including such diverse topics as the spread of rabies across Europe, the sprawl of urban development and the clumping of heather. One of the main tools available for analysis of spatial variation is spatial autocorrelation, and this text pres- ents a cohesive and comprehensive account of both its theory and its application. A great deal of progress has taken place in the eight years since the authors first published Spatial Autocorrelation, and this book is in essence an extended revision.

an der HEIDEN, UWE. Analysis of Neural Networks. Lecture Notes in Biomathematics, Vol. 35. Springer- Verlag, Berlin, 1980, 157 pp. £13.00.

In this monograph the author sets up a fairly general system of nonlinear integral equations to model the relationship of spike activity and action potential in a neuron. The model includes the inhibitory-excitatory effects of neurons on each other in systems with possibly very large numbers of neurons. The treat- ment is sufficiently general to encompass several spe- cial cases that have previously been studied.

Despite references to a number of concrete exam- ples, this book consists mainly of classical functional analysis. It is primarily concerned with obtaining condi- tions on the functions in the model which ensure the existence of stationary solutions, whether they are unique and whether they are stable. The treatment has no statistical or stochastic elements, so that the book has little of interest to statisticians as such.

A. G. HAWKES Department of Statistics,

University College of Swansea, Swansea, Wales.

MOHSENIN, N. N. Thermal Properties of Foods and Agricultural Materials. Gordon and Breach, New York, 1980, 407 pp. $92.00.

This is a comprehensive review of experiments and techniques to establish relafionships describing the heating and cooling of a wide range of materials, and their effects. The six chapters cover basic concepts and theory, determination of specific heat and other ther- mal properties, and application to practical problems, with many numerical examples. Regression, linear and nonlinear, is widely used, as are graphical methods, mainly for predicting the effects of conditions such as temperature and moisture. The use of nondimensional groups to reduce the number of predictor variates is a recurring feature. The Appendix, containing tables, could be a useful source of data for examples on regression and curve-fitting.

C. R. BAINES Long Ashton Research Station,

Bristol, England.

CLIFF, A. D. and ORD, J. K. Spatial Processes, Models and Applications. Methuen, Andover, Hamp- shire, England, 1981, 266 pp. £12.50.

Interest in the analysis of spatial processes has grown rapidly, with studies including such diverse topics as the spread of rabies across Europe, the sprawl of urban development and the clumping of heather. One of the main tools available for analysis of spatial variation is spatial autocorrelation, and this text pres- ents a cohesive and comprehensive account of both its theory and its application. A great deal of progress has taken place in the eight years since the authors first published Spatial Autocorrelation, and this book is in essence an extended revision.

an der HEIDEN, UWE. Analysis of Neural Networks. Lecture Notes in Biomathematics, Vol. 35. Springer- Verlag, Berlin, 1980, 157 pp. £13.00.

In this monograph the author sets up a fairly general system of nonlinear integral equations to model the relationship of spike activity and action potential in a neuron. The model includes the inhibitory-excitatory effects of neurons on each other in systems with possibly very large numbers of neurons. The treat- ment is sufficiently general to encompass several spe- cial cases that have previously been studied.

Despite references to a number of concrete exam- ples, this book consists mainly of classical functional analysis. It is primarily concerned with obtaining condi- tions on the functions in the model which ensure the existence of stationary solutions, whether they are unique and whether they are stable. The treatment has no statistical or stochastic elements, so that the book has little of interest to statisticians as such.

A. G. HAWKES Department of Statistics,

University College of Swansea, Swansea, Wales.

MOHSENIN, N. N. Thermal Properties of Foods and Agricultural Materials. Gordon and Breach, New York, 1980, 407 pp. $92.00.

This is a comprehensive review of experiments and techniques to establish relafionships describing the heating and cooling of a wide range of materials, and their effects. The six chapters cover basic concepts and theory, determination of specific heat and other ther- mal properties, and application to practical problems, with many numerical examples. Regression, linear and nonlinear, is widely used, as are graphical methods, mainly for predicting the effects of conditions such as temperature and moisture. The use of nondimensional groups to reduce the number of predictor variates is a recurring feature. The Appendix, containing tables, could be a useful source of data for examples on regression and curve-fitting.

C. R. BAINES Long Ashton Research Station,

Bristol, England.

CLIFF, A. D. and ORD, J. K. Spatial Processes, Models and Applications. Methuen, Andover, Hamp- shire, England, 1981, 266 pp. £12.50.

Interest in the analysis of spatial processes has grown rapidly, with studies including such diverse topics as the spread of rabies across Europe, the sprawl of urban development and the clumping of heather. One of the main tools available for analysis of spatial variation is spatial autocorrelation, and this text pres- ents a cohesive and comprehensive account of both its theory and its application. A great deal of progress has taken place in the eight years since the authors first published Spatial Autocorrelation, and this book is in essence an extended revision.

an der HEIDEN, UWE. Analysis of Neural Networks. Lecture Notes in Biomathematics, Vol. 35. Springer- Verlag, Berlin, 1980, 157 pp. £13.00.

In this monograph the author sets up a fairly general system of nonlinear integral equations to model the relationship of spike activity and action potential in a neuron. The model includes the inhibitory-excitatory effects of neurons on each other in systems with possibly very large numbers of neurons. The treat- ment is sufficiently general to encompass several spe- cial cases that have previously been studied.

Despite references to a number of concrete exam- ples, this book consists mainly of classical functional analysis. It is primarily concerned with obtaining condi- tions on the functions in the model which ensure the existence of stationary solutions, whether they are unique and whether they are stable. The treatment has no statistical or stochastic elements, so that the book has little of interest to statisticians as such.

A. G. HAWKES Department of Statistics,

University College of Swansea, Swansea, Wales.

MOHSENIN, N. N. Thermal Properties of Foods and Agricultural Materials. Gordon and Breach, New York, 1980, 407 pp. $92.00.

This is a comprehensive review of experiments and techniques to establish relafionships describing the heating and cooling of a wide range of materials, and their effects. The six chapters cover basic concepts and theory, determination of specific heat and other ther- mal properties, and application to practical problems, with many numerical examples. Regression, linear and nonlinear, is widely used, as are graphical methods, mainly for predicting the effects of conditions such as temperature and moisture. The use of nondimensional groups to reduce the number of predictor variates is a recurring feature. The Appendix, containing tables, could be a useful source of data for examples on regression and curve-fitting.

C. R. BAINES Long Ashton Research Station,

Bristol, England.

CLIFF, A. D. and ORD, J. K. Spatial Processes, Models and Applications. Methuen, Andover, Hamp- shire, England, 1981, 266 pp. £12.50.

Interest in the analysis of spatial processes has grown rapidly, with studies including such diverse topics as the spread of rabies across Europe, the sprawl of urban development and the clumping of heather. One of the main tools available for analysis of spatial variation is spatial autocorrelation, and this text pres- ents a cohesive and comprehensive account of both its theory and its application. A great deal of progress has taken place in the eight years since the authors first published Spatial Autocorrelation, and this book is in essence an extended revision.

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