Graphs and Questionnaires, Volume 32 (North-Holland Mathematics Studies)
| Date: 28 April 2011, 05:29
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Graphs and Questionnaires, Volume 32 (North-Holland Mathematics Studies)
By C.F.Picard
* Publisher: North Holland
* Number Of Pages: 446
* Publication Date: 1980-01-15
* ISBN-10 / ASIN: 0444852395
* ISBN-13 / EAN: 9780444852397
PREFACE
It gives me great pleasure to see the publication now of the
English version of my work on graphs and questionnaires. As
there are already a great many execllent texts on graph theory
in the English language, it has seemed appropriate to shift
the emphasis more firmly onto the subject of questionnaires by
reducing the graph theoretical content from the first volume of
the French edition to three chapters. There we set out those
elements of graph theory essential for the development of the
mathematical techniques used in the theory of questionnaires.
Chapter 3 is devoted to a discussion of several operations on
graphs which, although they have appeared in many separate
publications, have not yet been given a systematic treatment
in any other book. Chapter 1 corresponds to chapters 1, 2 and
3 of the French edition, chapter 2 to chapters 4 and 5 and
chapter 3 to chapter 6 and chapter 7. The reduction has been
achieved by leaving out certain sections for which no summary
is given. The remaining topics are carried over without
modification.
Chapters 4 to 10 constitute the translation of the second
volume of the French edition.
Following the bibliography arranged by chapter, a supplementary
list of papers on the subject of questionnaires rounds off the
work.
Except as indicated by the above remarks, the account of the
contents given in the Preface to the French edition naturally
still applies.
The topics dealt with in this book have aroused the interest of
several authors who have been able to make original contributions
to the theory and to guide young postgraduate students
into this line of research.
This translation is leaving the presses at just the right time
and I hope that a favourable reception by research workers,
engineers and technicians will facilitate further progress to
new extensions and applications.
I should like to thank the translators who have often contributed
appreciable improvements as compared with the French text
and have made every effort to spot any errors.
I am also grateful to the North-Holland Publishing Company for
the care they have taken over the preparation and presentation
of the book.
CONTENTS
Preface
Preface to the French edition
Chapter I Fundamental properties of graphs
Exercises
Ordered pairs and product sets
The graph concept
Elementary operations and transitive
closures
Connectivity, equivalence and preorder
Graph representations
Various definitions of graphs
Graph isomorphisms
Adjacency matrices
The incidence matrix of a graph
Computer representation of graphs
Valuations
Coding
Paths, circuits and cocircuits
Chains and concatenation
Cocircuits, cocycles and cycles
Chapter II Latticoids and arborescences
Exercises
Circuitless graphs
Arborescences and trees
Arborescences and data processing
Simplexes and arborescences
Monoids and arborescences
Chains, paths and arborescent
procedures
Coding
Finite and infinite graphs
Transportation networks
Chapter III Operations on graphs
Exercises
General definitions
Unary operations
Transformations
Cartesian operations
Product and sum
Classes of vertices
Connectivities
Valuations
Latticoid operations
Chapter IV General properties of questionnaires
Exercises
Preliminaries
The concept of a questionnaire
Axioms and definitions
Cutsets of a questionnaire
Partitions of the answers
Probabilities in an arborescent
questionnaire
Routing
The arborescence of paths in a latticoid
questionnaire
Compatible arborescent questionnaires
Probabilities in a latticoid
questionnaire
Probabilities of the vertices
Probabilities of the arcs and
conditioning
Example of a semantic
A restriction of the theory
Routing length
Chapter V The construction of questionnaires
Exercises
Operations on questionnaires
Definitions
Operations and routing length
Valuations on the answers and the arcs
L-optimal supports
Homogeneous questionnaires
a-I is a divisor of N-l
a-I is not a divisor of N-l
Heterogeneous questionnaires
Properties of arborescent questionnaires
The number of vertices and notation
Arborescences of minimal height
Questionnaires with balances support
Arborescences and questionnaires of
maximal height
Extremal properties of the supports
Chapter VI Optimal routing
Determination of an L-optimal
questionnaire
Necessary conditions for L-optimality
Substitutions of arcs
Transfers of arborescences
Sub-questionnaires
A sufficient condition for L-optimality
Huffman's algorithm
Questionnaires and coding
Equiprobable polychotomic questionnaires
Exercises
A characteristic property of homogeneous
balanced arborescences
Equiprobable dichotomic questionnaires
Optimal questionnaires
Routing in a dichotomic questionnaire
which is not optimal
Chapter VII Informational study of questionnaires
Exercises
Introduction to information
Hartley and Shannon's forms
Questionnaires in the sense of Shannon
Axiomatics of information
Faddeev's axioms
Some axiom systems
Properties of information
Convexity and concavity
Independence and dependence
Processed information and transmitted
information
Other definitions of information
Information for incomplete distributions
Measure, probability and information
Probability and information
Information for measure spaces
Non-probabilistic questionnaires
Chapter VIII Information and routing length
Information and routing in questionnaires
Inefficiency and noise
L-optimal questionnaires (LH ~ I)
Questionnaire product of two polychotomic
questionnaires
Heterogeneous questionnaires
Contribution of information
Maximization of processed information
and Shannon-Fano's algorithm
Partitions in equiprobable dichotomic
questionnaires and choice
Minimization of the contributed
information and Huffman's algorithm
Dichotomic questionnaires
Polychotomic questionnaires in the strict
sense
Heterogeneous questionnaires
Polychotomic questionnaires in the broad
sense
Informational interpretation of Huffman's
algorithm
Heterogeneous information and acquisition
Quasi-questionnaires
Quasi-answers and quasi-questions
Exercises
Instantaneous codes
Upper bounds for L-optimal questionnaires
Chapter IX Conditioning of the questions and answers
Limitations and extensions
Utilities of the answers
Useful length
Useful information
Cost of the questions
Costs and expenses
Free costs
Binding of the costs to the bases
Logarithmic costs
Questionnaires in the sense of Campbell
Questionnaires and Renyi's information
Charges and expenses
Questionnaires in the broad sense
Infinite questionnaires
Questionnaires with circuits
Flow charts and circuits
Realizable questionnaires
Constraints in questionnaires
A detection problem
Partitions and formable questions
Arborescent realizable L-optimal
questionnaires
The equivalence of constraints and costs
Questionnaires in practice
The dynamic aspect of interrogation
A random experiment
Absorption tests
Weighings
Interrogations, comparisons, sortings
Indirect interrogations and pseudoquestionnaires
Direct and indirect interrogations
Pseudoquestionnaires
Routing, information, convergence
Applications to pattern recognition
Diagnosis aid
Segmentation in a population
Word recognition
Comparisons and questions
Compatible realizable latticoids and
arborescences
Products of arborescent questionnaires
Sequential questionnaires
Questionnaires for sorting
L-optimal sorting
Realizable sortings
Problems
Solutions to Problems
Tables
Bibliography
Index
Main Symbols
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