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Design theory
Discrete Mathematics 73 (1989) 313-320 North-Holland
BOOK ANNOUNCEMENTS DESIGN THEORY Prof Dr. Thomas BETH, Prof Dr. Dieter JUNGNICKEL, Prof Dr. Hanfried LEN2 Cambridge University l%ss, Cambridge, London, New York, New Rochelle, Melbourne, Sydney. 1987, 688pages. $79.58. CONTENTS I. Exampbs and basic definitions
Incidence structures and incidence matrices Block designs and examples from afgne and projective geometry t-designs, Steiner systems and configurations Isomorphisms, duality, and correlations Partitions of the block set and resolvability Groop divisible incidence structures Transversal designs and nets Subspaces Hadamard designs II. Combiitorial
amdysis of designs
Basics Fisher’s inequality for painvise balanced designs Symmetric designs Symmetric designs continued: the theorem of Bruck, Chowla, and Ryser Balanced incidence structures with balanced duals More counting: the generalisation of Fisher’s inequality Extensions of designs Afhne designs The Hall-Connor theorem III.
Groups and designs Introduction Incidence morphisms Permutation groups Applications to incidence structures Examples from classical geometry Constructions of t-designs from groups Extensions of groups Construction of t-designs from base blocks
IV. Witt designs and Mathieu groups The existence of the Witt designs
The uniqueness of the small Witt designs The little Mathieu groups Properties of the large Witt design 5(5,8; 24) Some simple groups Witt’s construction of the Mathieu groups and Witt designs Hussain structures and the uniqueness of $(3,6; 12) The H&man-Sims group
313
Book announcements
314
V. HighIy transitive groups Sharply t-transitive groups z-homogeneous groups Concluding remarks: t-transitive groups VI. Diierence sets stud regdar symmetric designs Basic facts Multipliers Group rings The multiplier theorem Further results on mdtipliers Planar and Hadamard difference sets Some non-cyclic difference sets VII.
DiBlerence families Definitions and notations Base blocks and difference families More examples Construction of triple systems Some difference families in Galois fields Blocks with evenly distributed differences Some more special block designs Proof of Wilson’s Theorem 6.6
VIU. Further direct coustrmtions Pure and mixed differences Applications to the construction of resolvable block designs A difference construction for transversal designs Different cycles Infinite t-designs Further constructions using difference cycles The existence of t-designs for large values of A More transversal designs Unitals, ovals, and Baer subplanes IX. Recursive constructions Product constructions Use of pairwise balanced designs Use of groop divisible designs Applications of Hanani’s lemmas Block designs of block size three and four Solution of Kirkman’s school girl problem The basis of a closed set Block designs with block size 5 Groop divisible designs with block size 3 and 4 Concluding remarks X. Transversal designs and nets, co&wed A recursive construction Transversal designs with A > 1 A construction of Wilson Six and more mutually orthogonal Latin Squares The theorem of Chowla, ErdCis, and Straus The Bose-Bush bound Bruck’s completion theorem
Book announcements
Maximal nets with large deficiency Translation nets and maximal nets with small deficiency Completion results for p> 1 Extending symmetric nets Complete mappings and difference matrices Tarry’s theorem Xl. Asymptotic existence theory
Preliminaries The existence of Steiner systems with t) in given residue classes The main theorem for Steiner systems S(2, k; u) The eventual periodic&y of closed sets The main theorem for A= 1 The main theorem for il > 1 An existence theorem for resolvable block designs XII.
Characterisationsof classical designs
Projective and affine spaces as linear spaces Characterisations of projective spaces Characterisation of afhne spaces Locally projective linear spaces Good blocks Concluding remarks Appendix. Tables
Block designs Symmetric designs Abelian difference sets Small Steiner systems Series of Steinter systems Remark on t-designs with t 3 3 Orthogonal Latin squares
APPLIED LINEAR ALGEBRA Riaz A. USMANI Marcel Dekker, hc.,
New York and Base& 1887, 258pages.
CONTENTS 1. Vector Spaces
Fields Vector Spaces Matrices Other Examples of Vector Spaces Subspaces Linear Dependence, Basis, and Dimension Linear Dependence of a Set of Functions Vector Norms and Inner Products 2. Matrices
Product of Two Matrices The Operation of Transg osition Inversion Inverses for Some Special Matrices: Band Matrices
315
316
Book announcements The Row and Column Spaces of a Matrix Nullspace of a Matrix, Elementary Matrices
3. Genera&d Inverses Right and Left Inverses Principle of Least Squares Matrix Methods for Least Squares Problems The Moore-Penrose Inverse 4. I.&e&r Trolasronrrptiob Definition and Examples The Matrix of a Linear Transformation Transition Matrix and Change of Basis 5. The Eigenvalue Problem Eigenvalues and Eigenvectors Eigenvalues of Some Special Matrices Spectral Theorem Jordan’s Canonical Form Positive Deli&e Matrices 6. Functions of a Mahis Introduction Functions of a Matrix Defined by a Power Series Evaluation of exp(ul) by the Laplace Transform Method 7. Irreducible and Monotone M&ices Irreducible Matrices Monotone Matrices Application to the Two-Point Linear Boundary Value Problem NonlinearTwo-Point Boundary Value Problem Detembumts Determinants Linear Equations
Appendixz
and Linear Equations
A PRIMER OF DISCRETE MATHEMATICS Daniel. T. FINICBEINER II, Wendell D. LINDSTROM W. H. Freeman and Co., New York, 1987,363pages. coNTElvrs
1.Fmulamental concepts Informal set theory Preliminary Remarks Sets, Elements, and Subsets An Algebra of Sets Counting the Elements of a Set The Inclusion-Exclusion Principle Methods of reasoning An Algebra of Statements Theorems and Direct Proofs Indirect Proofs The Natural Numbers Mathematical Induction
Book announcements
317
2. Relations, functions, and operations Basic concepts
Binary Relations Representations of Binary Relations Elements of Matrix Calculations Equivalence Relations and Partitions Partial Order Relations Functions and Binary Operations A Recapitulation
Mathematical Structures 3. Combiitorial
mathematics Basic methods of counting
Combinatorial Problems Arrangements of Distinct Objects Selections from Distinct Objects Selections and Arrangements of Nondistinct Objects Selections and Arrangements with Repetition Distributions: Assignments Distributions: Arrangements Further counting techniques
Recursive Counting The Concept of a Generating Function Computation with Generating Functions 4. Graph theory and applications Concepts and Terminology Recognizing Different Graphs Planar Graphs Euler Trails and Circuits Hamilton Paths and Circuits Uses of Colors in Graphs Trees Search Strategies and Mazes 5. Graphs: applications and algorithms
Weighted Graphs Matching Constructing a Maximal Matching Network Flow Constructing a Maximal Flow COMBINATORIAL THEORY AND STATISTICAL DESIGN Gregory M. CONSTANTINE
John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1987, 469 pages, $48.60 CONTENTS Though this be madness, yet there is method in?.
1.Ways to clwse The Essentials of Counting Occupancy
Hamlet, W. SHAKESPEARE
Book announcements
318 More on Counting Exercises Historical Note References
2. Generatlng tanctions The Formal Power Series The Combinatorial Meaning of Convolution Exercises Generating Functions for Stirling Numbers Bell Polynomials Recurrence Relations Exercises The Generating Function of Labeled Spanning Trees Exercises Partitions of an Integer Exercises A Generating Function for Solutions of Diophantine Systems in Nonnegative Integers Historical Note References 3. chssicd InverSion Inversion in the Vector Space of Polynomials Taylor Expansions Exercises Formal Laurent Series Multivariate Laurent Series Exercises The Ordinary Generating Function Exercises Gaussian Polynomials Exercises Notes References 4. Graphs Cycles, Trails, and Complete Subgraphs Exercises Strongly Regular Graphs Exercises Spectra, Walks, and Oriented Cycles Exercises Graphs with Extreme Spectra Notes References 5.
ows ln Networks Extremal Points of Convex Polyhedra Exercises Matching and Marriage Problems Exercises The Arc Coloring Lemmas Flows and Cuts Related Results
Book announcements Exercises The “Out of Kiiter” Method Matroids and the Greedy Algorithm Exercises Notes References 6. Counting in the Presemceof a Group The General Theory Recipe for P6lya’s Theorem Examples Following the Recipe The Cycle Index Exercises More Theory Recipe for DeBruijn’s Result Example Following the Recipe Exercises Notes References 7. Block Designs The Basic Structure of t-Designs Constructions of t-Designs Fisher’s Inequality Exercises Extending Symmetric Designs On the Existence of Symmetric Designs Automorphisms of Designs Exercises Association Schemes Exercises Notes References 8. Statistical Designs Random Variables Exercises Factorial Experiments Introducing the Model Exercises Blocking Exercises Mixed Factorials Exercises Notes References 9. Miibins Inversion The Mobius Function Miibius Inversion on Special Partially Ordered Sets Results of Weisner and Hall Exercises Counting with the Mobius Function Surjective Morphisms Exercises
319
320
Book announcements
Notes References Appemlix 1.Fmite Groups Notation and Definitions Basic Results Normal Subgroups and Factor Groups Sylow Subgroups Groups of Prime Power Order References Appendix 2. FWte Fields, Vector Spaces, and Finite Geometries Finite Fields Vector Spaces Finite Geometries References Appendix 3. ‘I&e Four Squares Theorem and WiiWsCancellation Law The Four Squares Theorem Witt’s Cancellation Law References
BOOK ANNOUNCEMENTS DESIGN THEORY Prof Dr. Thomas BETH, Prof Dr. Dieter JUNGNICKEL, Prof Dr. Hanfried LEN2 Cambridge University l%ss, Cambridge, London, New York, New Rochelle, Melbourne, Sydney. 1987, 688pages. $79.58. CONTENTS I. Exampbs and basic definitions
Incidence structures and incidence matrices Block designs and examples from afgne and projective geometry t-designs, Steiner systems and configurations Isomorphisms, duality, and correlations Partitions of the block set and resolvability Groop divisible incidence structures Transversal designs and nets Subspaces Hadamard designs II. Combiitorial
amdysis of designs
Basics Fisher’s inequality for painvise balanced designs Symmetric designs Symmetric designs continued: the theorem of Bruck, Chowla, and Ryser Balanced incidence structures with balanced duals More counting: the generalisation of Fisher’s inequality Extensions of designs Afhne designs The Hall-Connor theorem III.
Groups and designs Introduction Incidence morphisms Permutation groups Applications to incidence structures Examples from classical geometry Constructions of t-designs from groups Extensions of groups Construction of t-designs from base blocks
IV. Witt designs and Mathieu groups The existence of the Witt designs
The uniqueness of the small Witt designs The little Mathieu groups Properties of the large Witt design 5(5,8; 24) Some simple groups Witt’s construction of the Mathieu groups and Witt designs Hussain structures and the uniqueness of $(3,6; 12) The H&man-Sims group
313
Book announcements
314
V. HighIy transitive groups Sharply t-transitive groups z-homogeneous groups Concluding remarks: t-transitive groups VI. Diierence sets stud regdar symmetric designs Basic facts Multipliers Group rings The multiplier theorem Further results on mdtipliers Planar and Hadamard difference sets Some non-cyclic difference sets VII.
DiBlerence families Definitions and notations Base blocks and difference families More examples Construction of triple systems Some difference families in Galois fields Blocks with evenly distributed differences Some more special block designs Proof of Wilson’s Theorem 6.6
VIU. Further direct coustrmtions Pure and mixed differences Applications to the construction of resolvable block designs A difference construction for transversal designs Different cycles Infinite t-designs Further constructions using difference cycles The existence of t-designs for large values of A More transversal designs Unitals, ovals, and Baer subplanes IX. Recursive constructions Product constructions Use of pairwise balanced designs Use of groop divisible designs Applications of Hanani’s lemmas Block designs of block size three and four Solution of Kirkman’s school girl problem The basis of a closed set Block designs with block size 5 Groop divisible designs with block size 3 and 4 Concluding remarks X. Transversal designs and nets, co&wed A recursive construction Transversal designs with A > 1 A construction of Wilson Six and more mutually orthogonal Latin Squares The theorem of Chowla, ErdCis, and Straus The Bose-Bush bound Bruck’s completion theorem
Book announcements
Maximal nets with large deficiency Translation nets and maximal nets with small deficiency Completion results for p> 1 Extending symmetric nets Complete mappings and difference matrices Tarry’s theorem Xl. Asymptotic existence theory
Preliminaries The existence of Steiner systems with t) in given residue classes The main theorem for Steiner systems S(2, k; u) The eventual periodic&y of closed sets The main theorem for A= 1 The main theorem for il > 1 An existence theorem for resolvable block designs XII.
Characterisationsof classical designs
Projective and affine spaces as linear spaces Characterisations of projective spaces Characterisation of afhne spaces Locally projective linear spaces Good blocks Concluding remarks Appendix. Tables
Block designs Symmetric designs Abelian difference sets Small Steiner systems Series of Steinter systems Remark on t-designs with t 3 3 Orthogonal Latin squares
APPLIED LINEAR ALGEBRA Riaz A. USMANI Marcel Dekker, hc.,
New York and Base& 1887, 258pages.
CONTENTS 1. Vector Spaces
Fields Vector Spaces Matrices Other Examples of Vector Spaces Subspaces Linear Dependence, Basis, and Dimension Linear Dependence of a Set of Functions Vector Norms and Inner Products 2. Matrices
Product of Two Matrices The Operation of Transg osition Inversion Inverses for Some Special Matrices: Band Matrices
315
316
Book announcements The Row and Column Spaces of a Matrix Nullspace of a Matrix, Elementary Matrices
3. Genera&d Inverses Right and Left Inverses Principle of Least Squares Matrix Methods for Least Squares Problems The Moore-Penrose Inverse 4. I.&e&r Trolasronrrptiob Definition and Examples The Matrix of a Linear Transformation Transition Matrix and Change of Basis 5. The Eigenvalue Problem Eigenvalues and Eigenvectors Eigenvalues of Some Special Matrices Spectral Theorem Jordan’s Canonical Form Positive Deli&e Matrices 6. Functions of a Mahis Introduction Functions of a Matrix Defined by a Power Series Evaluation of exp(ul) by the Laplace Transform Method 7. Irreducible and Monotone M&ices Irreducible Matrices Monotone Matrices Application to the Two-Point Linear Boundary Value Problem NonlinearTwo-Point Boundary Value Problem Detembumts Determinants Linear Equations
Appendixz
and Linear Equations
A PRIMER OF DISCRETE MATHEMATICS Daniel. T. FINICBEINER II, Wendell D. LINDSTROM W. H. Freeman and Co., New York, 1987,363pages. coNTElvrs
1.Fmulamental concepts Informal set theory Preliminary Remarks Sets, Elements, and Subsets An Algebra of Sets Counting the Elements of a Set The Inclusion-Exclusion Principle Methods of reasoning An Algebra of Statements Theorems and Direct Proofs Indirect Proofs The Natural Numbers Mathematical Induction
Book announcements
317
2. Relations, functions, and operations Basic concepts
Binary Relations Representations of Binary Relations Elements of Matrix Calculations Equivalence Relations and Partitions Partial Order Relations Functions and Binary Operations A Recapitulation
Mathematical Structures 3. Combiitorial
mathematics Basic methods of counting
Combinatorial Problems Arrangements of Distinct Objects Selections from Distinct Objects Selections and Arrangements of Nondistinct Objects Selections and Arrangements with Repetition Distributions: Assignments Distributions: Arrangements Further counting techniques
Recursive Counting The Concept of a Generating Function Computation with Generating Functions 4. Graph theory and applications Concepts and Terminology Recognizing Different Graphs Planar Graphs Euler Trails and Circuits Hamilton Paths and Circuits Uses of Colors in Graphs Trees Search Strategies and Mazes 5. Graphs: applications and algorithms
Weighted Graphs Matching Constructing a Maximal Matching Network Flow Constructing a Maximal Flow COMBINATORIAL THEORY AND STATISTICAL DESIGN Gregory M. CONSTANTINE
John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore, 1987, 469 pages, $48.60 CONTENTS Though this be madness, yet there is method in?.
1.Ways to clwse The Essentials of Counting Occupancy
Hamlet, W. SHAKESPEARE
Book announcements
318 More on Counting Exercises Historical Note References
2. Generatlng tanctions The Formal Power Series The Combinatorial Meaning of Convolution Exercises Generating Functions for Stirling Numbers Bell Polynomials Recurrence Relations Exercises The Generating Function of Labeled Spanning Trees Exercises Partitions of an Integer Exercises A Generating Function for Solutions of Diophantine Systems in Nonnegative Integers Historical Note References 3. chssicd InverSion Inversion in the Vector Space of Polynomials Taylor Expansions Exercises Formal Laurent Series Multivariate Laurent Series Exercises The Ordinary Generating Function Exercises Gaussian Polynomials Exercises Notes References 4. Graphs Cycles, Trails, and Complete Subgraphs Exercises Strongly Regular Graphs Exercises Spectra, Walks, and Oriented Cycles Exercises Graphs with Extreme Spectra Notes References 5.
ows ln Networks Extremal Points of Convex Polyhedra Exercises Matching and Marriage Problems Exercises The Arc Coloring Lemmas Flows and Cuts Related Results
Book announcements Exercises The “Out of Kiiter” Method Matroids and the Greedy Algorithm Exercises Notes References 6. Counting in the Presemceof a Group The General Theory Recipe for P6lya’s Theorem Examples Following the Recipe The Cycle Index Exercises More Theory Recipe for DeBruijn’s Result Example Following the Recipe Exercises Notes References 7. Block Designs The Basic Structure of t-Designs Constructions of t-Designs Fisher’s Inequality Exercises Extending Symmetric Designs On the Existence of Symmetric Designs Automorphisms of Designs Exercises Association Schemes Exercises Notes References 8. Statistical Designs Random Variables Exercises Factorial Experiments Introducing the Model Exercises Blocking Exercises Mixed Factorials Exercises Notes References 9. Miibins Inversion The Mobius Function Miibius Inversion on Special Partially Ordered Sets Results of Weisner and Hall Exercises Counting with the Mobius Function Surjective Morphisms Exercises
319
320
Book announcements
Notes References Appemlix 1.Fmite Groups Notation and Definitions Basic Results Normal Subgroups and Factor Groups Sylow Subgroups Groups of Prime Power Order References Appendix 2. FWte Fields, Vector Spaces, and Finite Geometries Finite Fields Vector Spaces Finite Geometries References Appendix 3. ‘I&e Four Squares Theorem and WiiWsCancellation Law The Four Squares Theorem Witt’s Cancellation Law References