**Editor for this issue:** Scott Fults <scottlinguistlist.org>

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- LINCOM EUROPA, H. M. Hubey, Mathematical Foundations of Linguistics

MATHEMATICAL FOUNDATIONS OF LINGUISTICS H. Mark Hubey, Montclair State University Only a few decades ago, only mathematicians, physicists and engineers took calculus courses, and calculus was tailored for them using examples from physics. This made it difficult for students from the life sciences including biology, economics, and psychology to learn mathematics. Recently books using examples from the life sciences and economics have become more popular for such students. Such a math book does not exist for linguists. Even the computational linguistics books (Formal Language Theory) are written for mathematicians and computer scientists. This book is for linguists. It is intended to teach the required math for a student to be a scientific linguist and to make linguistics a science on par with economics, and computer science. There are many concepts that are central to the sciences. Most students never see these in one place and if they do, they have to wait until graduate school to obtain them in the often-dreaded "quantitative" courses. As a result sometimes it takes years or even decades before learners are able to integrate what they have learned into a whole, if ever. We have little time and much to do. In addition to all of these problems we are now awash in data and information. It is now that the general public should be made aware of the solution to all of these problems. The answer is obviously "knowledge compression". Knowledge is structured information; it is a system not merely a collection of interesting facts. What this book does, and what all other math books do is teach people the tools with which they can structure and thus compress information and knowledge around them. It has also been said that mathematics is the science of patterns; it is exactly by finding such patterns that we compress knowledge. We can say that mathematics is the science of knowledge compression or information compression. This book provides the basic tools for mathematics (even including a short and intuitive explanation of differential and integral calculus). The broad areas of linguistics, probability theory, speech synthesis, speech recognition, computational linguistics (formal languages and machines), historical linguistics require mathematics of counting/combinatorics, Bayesian theory, correlation-regression analysis, stochastic processes, differential equations, vectors/tensors. These in turn are based on set theory, logic, measurement theory, graph theory, algebra, Boolean algebra, harmonic analysis etc. The mathematical fields introduced here are all common ideas from one which one can branch off into more advanced study in any of these fields thus this book brings together ideas from many disparate fields of mathematics which would not normally be put together into a single course. This is what makes this a book especially written for linguists. Table of Contents: 1. Generic Building Blocks Layering Numerals, Multiplication, Constants and Variables Summation--Gauss Zeno's Paradox and Euler Continuous Products Decision Trees, Prisoner's Dilemma CPM/PERT Methods 2. Symbolic Computation, Iteration and Recursion Algorithmic Definition of Integers Parallel and Serial Choices Multiplicative vs. Additive Intelligence Strong vs. Long Chain Trade-off Recursion/Iteration and Solution of a Nonlinear Equation Programming Charts Learning Iteration Frequency vs. Wavelength 3. Basic Counting and Reasoning Principles Product, Series Logical-AND Rule 4. Hazards of Doing Science Dimensionless Numbers Mass vs. Surface Area 5. Normalization Grade Normalization, Boxing Normalization Extensive vs. Intensive Variables Gymnastics & Diving Boyle's Law and Charles's Law Color Space & Vectors Torque Brain and Body Mass 6. Accuracy and Precision Significant Digits Paleontology 7. Reliability and Validity Ratio Scale Distance Hamming Distance Phonological Distance -- Distinctive Features Vowels and Consonants- Ordinal Cube What's a Bird? Interval Scale Temperature Scale Ordinal Scale Likert Scale Nominal Scale Sets, and Categorization 8. Sets: An Introduction Languages Cardinality, Empty Set Union, Intersection, Partition, Power Set, Complement, Difference Characteristic Bitstrings (functions) 9. Graphs: An Introduction Subgraphs, Unions & Intersections of Graphs Graph Representation: Incidence, Adjacency, Degree, Paths, Digraphs Hypercubes, Complete Graphs, Bipartite Graphs Multiple Comparisons of Historical Linguistics Representation: Incidence and Adjacency Matrices Euler Circuits 10. Objects & Spaces Cartesian Products, Vectors, Matrices, Tensors Matrix Multiplication Zero-One Matrices, Toeplitz matrices Markov Matrices, Leontieff Matrices, Phonotactics Matrices Rotation Matrices of Computer Graphics Sonority Scale and Vectors Venn Diagrams (Set Independence?) 11. Algebra: How many kinds are there? Arithmetic Language Capability Substitutes and complements Intelligence Theory and Testing 12. Boolean algebra Infinity Arithmetic Electrical Circuits and Infinity Parallel Circuits vs. Series Circuits XOR, EQ Representation of Integers Hamming Distance and XOR Phoneme Maps 13. Propositional Logic Implication Hempel's Raven paradox Paradoxes of Logic Rules of Inference Fallacies Integers: Division Algorithm Divisibility Fundamental Theorem of Arithmetic gcd, and lcm Mod, Div, and All that (methods of proof) Congruence Mod m Pseudo-Random Number Generators Caesar Cipher, ROT13, Comparative Method Fuzzy Logic Appendix: Axiomatizations of Logic 14. Quantification Syllogisms Continuous Products & Continuous Sums Predicates Quantification of Two Variables Mathematical Induction Time-Space Super-Liar Paradox 15. Relations Reflexive, Symmetric, Anti-symmetric, Transitive relations Representation of Relations Set-theoretic representation Matrix-representation Graph-theoretic representation 16. Boolean Matrices and Relations Composition of Relations--associative operation Powers of Matrices of Relations Equivalence Relation Inverses Operators and Operands (see Section 26: Operator Theory) 17. Partially Ordered Sets: partitions Hasse Diagrams -- prerequisite structure of this book Lattices, Subsets 18. Functions, Graphs, Vectors One-to-One Functions Onto Functions One-to-one correspondence Function Inverse Graph Isomorphism Minimal Spanning Trees Family Trees Cladistics Genetic Tree of Indo-European Languages and Isoglosses Vector Functions Distances on Vectors, Weighted Distances Intelligence Measurement Systems of Equations -- Algebraic Modeling 19. Asymptotic Analysis and Limits Big-O notation 20. Fuzzy Logic Axioms Invariants of Logic Continuous Logics Generalized Idempotent and Continuous Max-Min Operators 21. Counting Principles Pigeonhole Principle Sound Changes Permutations Combinations Words, Subsets, Sentences, Constrained Sentences Queues, Books, Phonotactics, Length constraints Distributing Objects to Containers Vervet Languages Distribution of Meanings Pascal's Identity, Vandermonde's Identity, Binomial Theorem Inclusion-Exclusion Theorem False Cognacy Problem 22. Induction, Recursion, Summation 23. Recurrence, Iteration, Counting Linear homogeneous first-order difference equation Fibonacci Series False Cognacy Problem Coupled Difference Equations Bitstrings and Polynomials Polya's Method of Counting 24. Formal Language Theory Real Human Languages Finite State Automata and Regular Languages Context-Free Languages Context-Sensitive Languages and Natural Languages 25. Simple Calculus Rates Integration from Summation 26. Probability Fundamentals Addition Theorem Multiplication Theorem Independence And Conditional Probability 27. Discrete Probability Theory from Counting 28. Bayes Theorem 29. Operator Theory (see Chapter 17) Linear Operators Commutativity Integration and Differentiation 30. Statistics Histogram Correlation-Regression 31. Expectation Operator & Density Functions Expectation and Moments 32. Discrete Probability Functions (Mass Functions) Uniform Geometric - Bernoulli Binomial Hypergeometric Poisson Birthday Problem 33. Continuous Probability Density Functions Uniform Exponential Gamma Density and Chi-Square Density Gaussian Density and the Central Limit Theorem 34. Joint and Marginal Density Functions 35. Stochastic Processes Stationarity Markov Processes Chapman-Kolmogorov Equations Speech Recognition Random Walk 36. Harmonic Analysis Delta function Fourier Series and Fourier Transform 37. Differential Equations, Green's Function and the Convolution Integral Complete solution of the First Order Linear DE Carbon Dating Menzerath's Law Altmann's Law Damped Harmonic Oscillator 38. Time and Ensemble Moments - Stationarity and Ergodicity Stationarity Ensemble Correlation Functions Time Averages and Ergodicity 39. Characteristic Functions, Moments and Cumulants 40. Stochastic Response of Linear Systems Example of Word Production in a Language Stochastic Excitation of the DHO 41. Fokker-Planck- Kolmogorov Methods Generalization of the Random Walk Replacement of a General Process by a Markov Process Appendices; Calculation of some integrals; References. ISBN 3 89586 641 5. LINCOM Handbooks in Linguistics 10. 260 pp. Pb: EUR 36.81 / USD 48 / DM 72 ISBN 3 89586 923 6. LINCOM Handbooks in Linguistics 10 (Hardcover). 260 pp. Hb: EUR 67.49 / USD 73 / DM 132 Ordering information for individuals: Please give us your creditcard no. / expiry date or send us a cheque. Prices in this information include shipment worldwide by airmail. A standing order for this series is available with special discounts offered to individual subscribers. LINCOM EUROPA, Paul-Preuss-Str. 25, D-80995 Muenchen, Germany; FAX +4989 3148909; http://home.t-online.de/home/LINCOM.EUROPA LINCOM.EUROPAt-online.de.Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

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