Permanents made their first appearance in 1812 in the famous memoirs of Binet and Cauchy. Since then 155 other mathematicians contributed 301 publications to the subject, more than three-quarters of which appeared in the last 19 years. The present monograph is an outcome of this remarkable re-awakening of interest in the permanent function.

The purpose of the book is to give a complete account of the theory of permanents, their history and applications, in a form accessible not only to mathematicians but also to workers in various applied fields, and to students of pure and applied mathematics. Here is the first complete account of the theory of permanents. It is a survey in the style of MacDuffy The Theory of Matrices and of A Survey of Matrix Theory and Matrix Inequalities, by Marcus and Minc. However, it differs from both works in several respects: the style is more leisurely, the proportion of theorems proved in the text is higher, and the scope is wider—the volume covers virtually the whole of the subject, a feature that no survey of the theory of matrices can even attempt. Apart from many theorems proved in detail, there are numerous results stated without proof. Due to limitation of space, not every known result could be mentioned in the text. The choice of the theorems included in the book reflects, of course, the author's predilections.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

The theory of partitions is one of the very few branches of mathematics that can be appreciated by anyone who is endowed with little more than a lively interest in the subject. Its applications are found wherever discrete objects are to be counted or classified, whether in the molecular and the atomic studies of matter, in the theory of numbers, or in combinatorial problems from all sources.

Professor Andrews has written the first thorough survey of this many sided field. The specialist will consult it for the more recondite results, the student will be challenged by many a deceptively simple fact, and the applied scientist may locate in it the missing identity he needs to organize his data.

Measurement

A major difference between a “well-developed” science such as physics and some of the less “well-developed” sciences such as psychology or sociology is the degree to which things are measured. In this volume, we develop a theory of measurement that can act as a foundation for measurement in the social and behavioral sciences. Starting with such classical measurement concepts of the physical sciences as temperature and mass, we extend a theory of measurement to the social sciences. We discuss the measurement of preference, loudness, brightness, intelligence, and so on. We also apply measurement to such societal problems as air and noise pollution, weather forecasting, and public health, and comment on the development of pollution indices and consumer price indices.

Throughout, we apply the results to decisionmaking. The decisionmaking applications deal with transportation, consumer behavior, environmental problems, energy use, and medicine, as well as with laboratory situations involving human and animal subjects.

In this introduction, we try to give the reader a quick preview of the contents and organization of the book, and of the problems we shall address. The reader might prefer to read the introduction rather quickly the first time, and to return to it later.

A large body of mathematics consists of facts that can be presented and described much like any other natural phenomenon. These facts, at times explicitly brought out as theorems, at other times concealed within a proof, make up most of the applications of mathematics, and are the most likely to survive changes of style and of interest.

This ENCYCLOPEDIA will attempt to present the factual body of all mathematics. Clarity of exposition, accessibility to the non-specialist, and a thorough bibliography are required of each author. Volumes will appear in no particular order, but will be organized into sections, each one comprising a recognizable branch of present-day mathematics. Numbers of volumes and sections will be reconsidered as times and needs change.

It is hoped that this enterprise will make mathematics more widely used where it is needed, and more accessible in fields in which it can be applied but where it has not yet penetrated because of insufficient information.

The Psychophysical Problem

Loudness

A sound has a variety of physical characteristics. For example, a pure tone can be described by its physical intensity (energy transported), its frequency (in cycles per second), its duration, and so on. The same sound has various psychological characteristics. For example, how loud does it seem? What emotional meaning does it portray? What images does it suggest? Since the middle of the nineteenth century, scientists have tried to study the relationships between the physical characteristics of stimuli like sounds and their psychological characteristics. Some psychological characteristics might have little relationship to physical ones. For example, emotional meaning probably has little relation to the physical intensity of a sound, but rather it may be related to past experiences, as, for example, with the sound of a siren. Other psychological characteristics seem to be related in fairly regular ways to physical characteristics. Such sensations as loudness are an example. Psychophysics is the discipline that studies various psychological sensations such as loudness, brightness, apparent length, and apparent duration, and their relations to physical stimuli. It attempts to scale or measure psychological sensations on the basis of corresponding physical stimuli. In this chapter, we shall describe some of the history of psychophysical scaling and its applications or potential applications to measurement of noise pollution, of attitudes, of utility, etc., and we shall discuss ways to put psychophysical scaling on a firm measurement-theoretic foundation. We shall concentrate on loudness.