Discrete mathematics Books

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Book SynopsisRichard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation and differential posets.Trade Review'… sure to become a standard as an introductory graduate text in combinatorics.' Bulletin of the American Mathematical Society'… will engage from start to finish the attention of any mathematician who will open it at page one.' Gian-Carlo RotaTable of Contents1. What is enumerative combinatorics?; 2. Sieve methods; 3. Partially ordered sets; 4. Rational generating functions.

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Book SynopsisThe projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes and graphs. This book provides an introduction to these geometries and their many applications to other areas of combinatorics. Coverage includes a detailed treatment of the forbidden subgraph problem from a geometrical point of view, and a chapter on maximum distance separable codes, which includes a proof that such codes over prime fields are short. The author also provides more than 100 exercises (complete with detailed solutions), which show the diversity of applications of finite fields and their geometries. Finite Geometry and Combinatorial Applications is ideal for anyone, from a third-year undergraduate to a researcher, who wishes to familiarise themselves with and gain an appreciation of finite geometry.Table of Contents1. Fields; 2. Vector spaces; 3. Forms; 4. Geometries; 5. Combinatorial applications; 6. The forbidden subgraph problem; 7. MDS codes; Appendix A. Solutions to the exercises; Appendix B. Additional proofs; Appendix C. Notes and references; References; Index.

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Book SynopsisFrom social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.Trade Review'This is a well-planned book that is true to its title in that it is indeed accessible for anyone with just an undergraduate student's knowledge of enumerative combinatorics and probability.' Miklós Bóna, MAA ReviewsTable of ContentsPreface; Part I. Basic Models: 1. Random graphs; 2. Evolution; 3. Vertex degrees; 4. Connectivity; 5. Small subgraphs; 6. Spanning subgraphs; 7. Extreme characteristics; 8. Extremal properties; Part II. Basic Model Extensions: 9. Inhomogeneous graphs; 10. Fixed degree sequence; 11. Intersection graphs; 12. Digraphs; 13. Hypergraphs; Part III. Other Models: 14. Trees; 15. Mappings; 16. k-out; 17. Real-world networks; 18. Weighted graphs; 19. Brief notes on uncovered topics; Part IV. Tools and Methods: 20. Moments; 21. Inequalities; 22. Differential equations method; 23. Branching processes; 24. Entropy; References; Author index; Main index.

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Book SynopsisCatalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their properties and applications to combinatorics, algebra, analysis, number theory, probability theory, geometry, topology, and other areas. Following an introduction to the basic properties of Catalan numbers, the book presents 214 different kinds of objects counted by them in the form of exercises with solutions. The reader can try solving the exercises or simply browse through them. Some 68 additional exercises with prescribed difficulty levels present various properties of Catalan numbers and related numbers, such as Fuss-Catalan numbers, Motzkin numbers, SchrÃder numbers, Narayana numbers, super Catalan numbers, q-Catalan numbers and (q,t)-Catalan numbers. The book ends with a history of Catalan numbers by Igor Pak and a glossary of key terms. Whether your interest in mathematics is recreation or research, you will find plenty of fTable of Contents1. Basic properties; 2. Bijective exercises; 3. Bijective solutions; 4. Additional problems; 5. Solutions to additional problems.

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Book SynopsisRichard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes more than 300 new exercises, many with solutions, updated and expanded chapter bibliographies and substantial new material on permutation statistics.Trade Review'… sure to become a standard as an introductory graduate text in combinatorics.' Bulletin of the American Mathematical Society'… will engage from start to finish the attention of any mathematician who will open it at page one.' Gian-Carlo RotaTable of Contents1. What is enumerative combinatorics?; 2. Sieve methods; 3. Partially ordered sets; 4. Rational generating functions.

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Book SynopsisTable of ContentsR. Algebra Reference R-1 Polynomials R-2 Factoring R-3 Rational Expressions R-4 Equations R-5 Inequalities R-6 Exponents R-7 Radicals 1. Linear Functions 1-1 Slopes and Equations of Lines 1-2 Linear Functions and Applications 1-3 The Least Squares Line Chapter Review Extended Application: Using Extrapolation to Predict Life Expectancy 2. Systems of Linear Equations and Matrices 2-1 Solution of Linear Systems by the Echelon Method 2-2 Solution of Linear Systems by the Gauss-Jordan Method 2-3 Addition and Subtraction of Matrices 2-4 Multiplication of Matrices 2-5 Matrix Inverses 2-6 Input-Output Models Chapter Review Extended Application: Contagion 3. Linear Programming: The Graphical Method 3-1 Graphing Linear Inequalities 3-2 Solving Linear Programming Problems Graphically 3-3 Applications of Linear Programming Chapter Review 4. Linear Programming: The Simplex Method 4-1 Slack Variables and the Pivot 4-2 Maximization Problems 4-3 Minimization Problems; Duality 4-4 Nonstandard Problems Chapter Review Extended Application: Using Integer Programming in the Stock-Cutting Problem 5. Mathematics of Finance 5-1 Simple and Compound Interest 5-2 Future Value of an Annuity 5-3 Present Value of an Annuity; Amortization Chapter Review Extended Application: Time, Money, and Polynomials 6. Logic 6-1 Statements 6-2 Truth Tables and Equivalent Statements 6-3 The Conditional and Circuits 6-4 More on the Conditional 6-5 Analyzing Arguments and Proofs 6-6 Analyzing Arguments with Quantifiers Chapter Review Extended Application: Logic Puzzles 7. Sets and Probability 7-1 Sets 7-2 Applications of Venn Diagrams 7-3 Introduction to Probability 7-4 Basic Concepts of Probability 7-5 Conditional Probability; Independent Events 7-6 Bayes' Theorem Chapter Review Extended Application: Medical Diagnosis 8. Counting Principles: Further Probability Topics 8-1 The Multiplication Principle; Permutations 8-2 Combinations 8-3 Probability Applications of Counting Principles 8-4 Binomial Probability 8-5 Probability Distributions; Expected Value Chapter Review Extended Application: Optimal Inventory for a Service Truck 9. Statistics 9-1 Frequency Distributions; Measures of Central Tendency 9-2 Measures of Variation 9-3 The Normal Distribution 9-4 Normal Approximation to the Binomial Distribution Chapter Review Extended Application: Statistics in the Law - The Castaneda Decision 10. Nonlinear Functions 10-1 Properties of Functions 10-2 Quadratic Functions; Translation and Reflection 10-3 Polynomial and Rational Functions 10-4 Exponential Functions 10-5 Logarithmic Functions 10-6 Applications: Growth and Decay; Mathematics of Finance Chapter Review Extended Application: Characteristics of the Monkeyface Prickleback 11. The Derivative 11-1 Limits 11-2 Continuity 11-3 Rates of Change 11-4 Definition of the Derivative 11-5 Graphical Differentiation Chapter Review Extended Application: A Model for Drugs Administered Intravenously 12. Calculating the Derivative 12-1 Techniques for Finding Derivatives 12-2 Derivatives of Products and Quotients 12-3 The Chain Rule 12-4 Derivatives of Exponential Functions 12-5 Derivatives of Logarithmic Functions Chapter Review Extended Application: Electric Potential and Electric Field 13. Graphs and the Derivative 13-1 Increasing and Decreasing Functions 13-2 Relative Extrema 13-3 Higher Derivatives, Concavity, and the Second Derivative Test 13-4 Curve Sketching Chapter Review Extended Application: A Drug Concentration Model for Orally Administered Medications (new) 14. Applications of the Derivative 14-1 Absolute Extrema 14-2 Applications of Extrema 14-3 Further Business Applications: Economic Lot Size; Economic Order Quantity; Elasticity of Demand 14-4 Implicit Differentiation 14-5 Related Rates 14-6 Differentials: Linear Approximation Chapter Review Extended Application: A Total Cost Model for a Training Program 15. Integration 15-1 Antiderivatives 15-2 Substitution 15-3 Area and the Definite Integral 15-4 The Fundamental Theorem of Calculus 15-5 The Area Between Two Curves 15-6 Numerical Integration Chapter Review Extended Application: Estimating Depletion Dates for Minerals 16. Further Techniques and Applications of Integration 16-1 Integration by Parts 16-2 Volume and Average Value 16-3 Continuous Money Flow 16-4 Improper Integrals 16-5 Solutions of Elementary and Separable Differential Equations* Chapter Review Extended Application: Estimating Learning Curves in Manufacturing with Integrals 17. Multivariable Calculus 17-1 Functions of Several Variables 17-2 Partial Derivatives 17-3 Maxima and Minima 17-4 Lagrange Multipliers 17-5 Total Differentials and Approximations 17-6 Double Integrals Chapter Review Extended Application: Using Multivariable Fitting to Create a Response Surface Design 18. Probability and Calculus 18-1 Continuous Probability Models 18-2 Expected Value and Variance of Continuous Random Variables 18-3 Special Probability Density Functions Chapter Review Extended Application: Exponential Waiting Times Tables Table 1 Formulas of/from Geometry Table 2 Area Under a Normal Curve Table 3 Integrals Answers to Selected Exercises Photo Acknowledgements Index

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Book SynopsisThe theory of graph coloring has existed for more than 150 years. This book states that in the case of hypergraphs, there exist problems on both the minimum and the maximum number of colors. This feature pervades the theory, methods, algorithms, and applications of mixed hypergraph coloring.Table of ContentsIntroduction The lower chromatic number of a hypergraph Mixed hypergraphs and the upper chromatic number Uncolorable mixed hypergraphs Uniquely colorable mixed hypergraphs $\mathcal{C}$-perfect mixed hypergraphs Gaps in the chromatic spectrum Interval mixed hypergraphs Pseudo-chordal mixed hypergraphs Circular mixed hypergraphs Planar mixed hypergraphs Coloring block designs as mixed hypergraphs Modelling with mixed hypergraphs Bibliography List of figures Index.

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Book Synopsis

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Book Synopsis

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Book SynopsisThis book aims to help engineers, Masters students and young researchers to understand and gain a general knowledge of logistic systems optimization problems and techniques, such as system design, layout, stock management, quality management, lot-sizing or scheduling. It summarizes the evaluation and optimization methods used to solve the most frequent problems. In particular, the authors also emphasize some recent and interesting scientific developments, as well as presenting some industrial applications and some solved instances from real-life cases.Performance evaluation tools (Petri nets, the Markov process, discrete event simulation, etc.) and optimization techniques (branch-and-bound, dynamic programming, genetic algorithms, ant colony optimization, etc.) are presented first. Then, new optimization methods are presented to solve systems design problems, layout problems and buffer-sizing optimization. Forecasting methods, inventory optimization, packing problems, lot-sizing quality management and scheduling are presented with examples in the final chapters.Trade Review“On the other hand, this book constitutes a valuable guide and convenient introduction to the fied of operations research applications for professionals, which deal with real production and logistic system design and management. It can be also recommended as a textbook for students of production management.” (Zentralblatt Math, 1 May 2013) Table of ContentsIntroduction xiii Chapter 1. Modeling and Performance Evaluation 1 1.1. Introduction 1 1.2. Markovian processes 2 1.2.1. Overview of stochastic processes 2 1.2.2. Markov processes 3 1.2.2.1. Basics 3 1.2.2.2. Chapman–Kolmogorov equations 4 1.2.2.3. Steady-state probabilities 5 1.2.2.4. Graph associated with a Markov process 6 1.2.2.5. Application to production systems 6 1.2.3. Markov chains 8 1.2.3.1. Basics 8 1.2.3.2. State probability vectors 9 1.2.3.3. Fundamental equation of a Markov chain 9 1.2.3.4. Graph associated with a Markov chain 10 1.2.3.5. Steady states of ergodic Markov chains 11 1.2.3.6. Application to production systems 12 1.3. Petri nets 14 1.3.1. Introduction to Petri nets 14 1.3.1.1. Basic definitions 14 1.3.1.2. Dynamics of Petri nets 15 1.3.1.3. Specific structures 16 1.3.1.4. Tools for Petri net analysis 18 1.3.1.5. Properties of Petri nets 19 1.3.2. Non-autonomous Petri nets 20 1.3.3. Timed Petri nets 20 vi Optimization of Logistics 1.3.4. Continuous Petri nets 23 1.3.4.1. Fundamental equation and performance analysis 24 1.3.4.2. Example 25 1.3.5. Colored Petri nets 27 1.3.6. Stochastic Petri nets 28 1.3.6.1. Firing time 29 1.3.6.2. Firing selection policy 29 1.3.6.3. Service policy 30 1.3.6.4. Memory policy 30 1.3.6.5. Petri net analysis 30 1.3.6.6. Marking graph 31 1.3.6.7. Generator of Markovian processes 31 1.3.6.8. Fundamental equation 32 1.3.6.9. Steady-state probabilities 32 1.3.6.10. Performance indices (steady state) 35 1.4. Discrete-event simulation 36 1.4.1. The role of simulation in logistics systems analysis 36 1.4.2. Components and dynamic evolution of systems 37 1.4.3. Representing chance and the Monte Carlo method 38 1.4.3.1. Uniform distribution U [0, 1] 38 1.4.3.2. The Monte Carlo method 39 1.4.4. Simulating probability distributions 41 1.4.4.1. Simulating random events 41 1.4.4.2. Simulating discrete random variables 44 1.4.4.3. Simulating continuous random variables 47 1.4.5. Discrete-event systems 52 1.4.5.1. Key aspects of simulation 52 1.5. Decomposition method 57 1.5.1. Presentation 57 1.5.2. Details of the method 58 Chapter 2. Optimization 61 2.1. Introduction 61 2.2. Polynomial problems and NP-hard problems 62 2.2.1. The complexity of an algorithm 62 2.2.2. Example of calculating the complexity of an algorithm 63 2.2.3. Some definitions 64 2.2.3.1. Polynomial-time algorithms 64 2.2.3.2. Pseudo-polynomial-time algorithms 64 2.2.3.3. Exponential-time algorithms 64 2.2.4. Complexity of a problem 64 2.2.4.1. Polynomial-time problems 64 2.2.4.2. NP-hard problems 64 2.3. Exact methods 64 2.3.1. Mathematical programming 64 2.3.2. Dynamic programming 65 2.3.3. Branch and bound algorithm 65 2.4. Approximate methods 66 2.4.1. Genetic algorithms 67 2.4.1.1. General principles 67 2.4.1.2. Encoding the solutions 67 2.4.1.3. Crossover operators 68 2.4.1.4. Mutation operators 70 2.4.1.5. Constructing the population in the next generation 70 2.4.1.6. Stopping condition 70 2.4.2. Ant colonies 70 2.4.2.1. General principle 70 2.4.2.2. Management of pheromones: example of the traveling salesman problem 71 2.4.3. Tabu search 72 2.4.3.1. Initial solution 73 2.4.3.2. Representing the solution 73 2.4.3.3. Creating the neighborhood 74 2.4.3.4. The tabu list 75 2.4.3.5. An illustrative example 76 2.4.4. Particle swarm algorithm 76 2.4.4.1. Description 76 2.4.4.2. An illustrative example 77 2.5. Multi-objective optimization 79 2.5.1. Definition 79 2.5.2. Resolution methods 80 2.5.3. Comparison criteria 81 2.5.3.1. The Riise distance 81 2.5.3.2. The Zitzler measure 82 2.5.4. Multi-objective optimization methods 82 2.5.4.1. Exact methods 82 2.5.4.2. Approximate methods 84 2.6. Simulation-based optimization 89 2.6.1. Dedicated tools 90 2.6.2. Specific methods 90 Chapter 3. Design and Layout 93 3.1. Introduction 93 3.2. The different types of production system 94 3.3. Equipment selection 97 viii Optimization of Logistics 3.3.1. General overview 97 3.3.2. Equipment selection with considerations of reliability 99 3.3.2.1. Introduction to reliability optimization 99 3.3.2.2. Design of a parallel-series system 100 3.4. Line balancing 110 3.4.1. The classification of line balancing problems 111 3.4.1.1. The simple assembly line balancing model (SALB) 111 3.4.1.2. The general assembly line balancing model (GALB) 112 3.4.2. Solution methods 112 3.4.2.1. Exact methods 112 3.4.2.2. Approximate methods 113 3.4.3. Literature review 113 3.4.4. Example 113 3.5. The problem of buffer sizing 114 3.5.1. General overview 116 3.5.2. Example of a multi-objective buffer sizing problem 116 3.5.3. Example of the use of genetic algorithms 117 3.5.3.1. Representation of the solutions 117 3.5.3.2. Calculation of the objective function 118 3.5.3.3. Selection of solutions for the archive 119 3.5.3.4. New population and stopping criterion 119 3.5.4. Example of the use of ant colony algorithms 119 3.5.4.1. Encoding 120 3.5.4.2. Construction of the ant trails 121 3.5.4.3. Calculation of the visibility 121 3.5.4.4. Global and local updates of the pheromones 122 3.5.5. Example of the use of simulation-based optimization 123 3.5.5.1. Simulation model 125 3.5.5.2. Optimization algorithms 129 3.5.5.3. The pairing of simulation and optimization 130 3.5.5.4. Results and comparison 130 3.6. Layout 132 3.6.1. Types of facility layout 132 3.6.1.1. Logical layout 132 3.6.1.2. Physical layout 133 3.6.2. Approach for treating a layout problem 133 3.6.2.1. Linear layout 134 3.6.2.2. Functional layout 135 3.6.2.3. Cellular layout 135 3.6.2.4. Fixed layout 135 3.6.3. The best-known methods 135 3.6.4. Example of arranging a maintenance facility 136 3.6.5. Example of laying out an automotive workshop 140 Chapter 4. Tactical Optimization 143 4.1. Introduction 143 4.2. Demand forecasting 143 4.2.1. Introduction 143 4.2.2. Categories and methods 144 4.2.3. Time series 145 4.2.4. Models and series analysis 146 4.2.4.1. Additive models 147 4.2.4.2. Multiplicative model 149 4.2.4.3. Exponential smoothing 150 4.3. Stock management 155 4.3.1. The different types of stocked products 156 4.3.2. The different types of stocks 157 4.3.3. Storage costs 157 4.3.4. Stock management 159 4.3.4.1. Functioning of a stock 159 4.3.4.2. Stock monitoring 161 4.3.4.3. Stock valuation 162 4.3.5. ABC classification method 163 4.3.6. Economic quantities 165 4.3.6.1. Economic quantity: the Wilson formula 166 4.3.6.2. Economic quantity with a discount threshold 167 4.3.6.3. Economic quantity with a uniform discount 168 4.3.6.4. Economic quantity with a progressive discount 169 4.3.6.5. Economic quantity with a variable ordering cost 170 4.3.6.6. Economic quantity with order consolidation 171 4.3.6.7. Economic quantity with a non-zero delivery time 172 4.3.6.8. Economic quantity with progressive input 172 4.3.6.9. Economic quantity with tolerated shortage 173 4.3.7. Replenishment methods 174 4.3.7.1. The (r, Q) replenishment method 175 4.3.7.2. The (T , S) replenishment method 175 4.3.7.3. The (s, S) replenishment method 175 4.3.7.4. The (T , r, S) replenishment method 176 4.3.7.5. The (T , r, Q) replenishment method 177 4.3.7.6. Security stock 177 4.4. Cutting and packing problems 178 4.4.1. Classifying cutting and packing problems 179 4.4.2. Packing problems in industrial systems 183 4.4.2.1. Model 183 4.4.2.2. Solution 185 4.5. Production and replenishment planning, lot-sizing methods 186 4.5.1. Introduction 186 x Optimization of Logistics 4.5.2. MRP and lot-sizing 186 4.5.3. Lot-sizing methods 187 4.5.3.1. The characteristic elements of the models 188 4.5.3.2. Lot-sizing in the scientific literature 189 4.5.4. Examples 190 4.5.4.1. The Wagner–Whitin method 191 4.5.4.2. The Florian and Klein method 193 4.6. Quality management 198 4.6.1. Evaluation, monitoring and improvement tools 198 4.6.1.1. The objective of metrology 198 4.6.1.2. Concepts of error and uncertainty 198 4.6.1.3. Statistical quality control 199 4.6.1.4. Stages of control 199 4.6.1.5. Tests of normality 200 4.6.2. Types of control 205 4.6.2.1. Reception or final control 205 4.6.2.2. Reception control by measurement 206 4.6.2.3. Manufacturing control 209 4.6.2.4. Control charts 214 Chapter 5. Scheduling 233 5.1. Introduction 233 5.2. Scheduling problems 234 5.2.1. Basic notions 234 5.2.2. Notation 234 5.2.3. Definition of the criteria and objective functions 234 5.2.3.1. Flow time 235 5.2.3.2. Lateness 235 5.2.3.3. Tardiness 235 5.2.3.4. The earliness 236 5.2.3.5. Objective functions 236 5.2.3.6. Properties of schedules 238 5.2.4. Project scheduling 239 5.2.4.1. Definition of a project 239 5.2.4.2. Projects with unlimited resources 240 5.2.4.3. Projects with consumable resources 247 5.2.4.4. Minimal-cost scheduling 252 5.2.5. Single-machine problems 254 5.2.5.1. Minimization of the mean flow time 5.2.5.2. Minimization of the mean weighted flow time 5.2.5.3. Minimization of the mean flow time 5.2.5.4. Minimization of the maximum tardiness Tmax, 1/ri = 0/Tmax 259 5.2.5.5. Minimization of the maximum tardiness when the jobs have different arrival dates, with pre-emption 1/ri, pmtn/Tmax 261 5.2.5.6. Minimization of the mean tardiness 1//T 261 5.2.5.7. Minimization of the flow time 1/ri/F 265 5.2.6. Scheduling a flow shop workshop 267 5.2.6.1. The two-machine problem 267 5.2.6.2. A particular case of the three-machine problem 268 5.2.6.3. The m-machine problem 268 5.2.7. Parallel-machine problems 270 5.2.7.1. Identical machines, ri = 0, M in F 270 5.2.7.2. Identical machines, ri = 0, M in Cmax interruptible jobs 271 Bibliography 273 Index 285

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