Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

/pdf/combinatorial-stochastic-processes-4nf5lxpd0t.pdf

Combinatorial Stochastic Processes

Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes. Generalized Fibonacci Numbers. Additional Fibonacci and Lucas Formulas. The Euclidean Algorithm. Solving Recurrence Relations. Completeness Theorems. Pascal's Triangle. Pascal-Like Triangles. Additional Pascal-Like Triangles. Hosoya's Triangle. Divisibility Properties. Generalized Fibonacci Numbers Revisited. Generating Functions. Generating Functions Revisited. The Golden Ratio. The Golden Ratio Revisited. Golden Triangles. Golden Rectangles. Fibonacci Geometry. Regular Pentagons. The Golden Ellipse and Hyperbola. Continued Fractions. Weighted Fibonacci and Lucas Sums. Weighted Fibonacci and Lucas Sums Revisited. The Knapsack Problem. Fibonacci Magic Squares. Fibonacci Matrices. Fibonacci Determinants. Fibonacci and Lucas Congruences. Fibonacci and Lucas Periodicity. Fibonacci and Lucas Series. Fibonacci Polynomials. Lucas Polynomials. Jacobsthal Polynomials. Zeros of Fibonacci and Lucas Polynomials. Morgan-Voyce Polynomials. Fibonometry. Fibonacci and Lucas Subscripts. Gaussian Fibonacci and Lucas Numbers. Analytic Extensions. Tribonacci Numbers. Tribonacci Polynomials. Appendix 1: Fundamentals. Appendix 2: The First 100 Fibonacci and Lucas Numbers. Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations. Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations. References. Solutions to Odd-Numbered Exercises. Index.

Fibonacci and Lucas Numbers with Applications

'Operads are powerful tools, and this is the book in which to read about them' - ""Bulletin of the London Mathematical Society"". Operads are mathematical devices that describe algebraic structures of many varieties and in various categories. Operads are particularly important in categories with a good notion of 'homotopy', where they play a key role in organizing hierarchies of higher homotopies. Significant examples from algebraic topology first appeared in the sixties, although the formal definition and appropriate generality were not forged until the seventies. In the nineties, a renaissance and further development of the theory were inspired by the discovery of new relationships with graph cohomology, representation theory, algebraic geometry, derived categories, Morse theory, symplectic and contact geometry, combinatorics, knot theory, moduli spaces, cyclic cohomology, and, last but not least, theoretical physics, especially string field theory and deformation quantization. The book contains a detailed and comprehensive historical introduction describing the development of operad theory from the initial period when it was a rather specialized tool in homotopy theory to the present when operads have a wide range of applications in algebra, topology, and mathematical physics. Many results and applications currently scattered in the literature are brought together here along with new results and insights. The basic definitions and constructions are carefully explained and include many details not found in any of the standard literature.

Operads in algebra, topology, and physics

The lattice of non-crossing partitions was introduced by Kreweras [Kre] in 1972 and examined further by Poupard [Pou] and Edelman [Edel, Ede2]. In the last time there has been an increased interest in this lattice, on one hand from the purely combinatorial point of view [SiU, EdS, Sire, BSS] and on the other hand from a (quantum) probabilistic point of view [GSS, Spel, Spe2, Spe3]. We became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures, the 'free' convolution introduced by Voiculescu [Voil, Voi2, Voi3]. This connection between the free convolution and the lattice of non-crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions. We shall work out this connection in the last part of this work. The intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other. Motivated by Voiculescu's formula for the R-series of free convolution we examine multiplicative functions on the lattice of non-crossing partitions and prove our main theorem, which is much in the spirit of Voiculescu's formula. We shall then use this main theorem for deriving some known results on non-crossing partitions in a unified way. This part of our work is purely combinatorial, the free convolution serves only as a motivation for our theorem. In the last section we shall switch to the probabilistie side and show how the lattice of non-crossing partitions determines the structure of the free convolution. Voiculescu's formula will then follow as an easy corollary of our main theorem. In this respect, we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools.

Multiplicative functions on the lattice of non-crossing partitions and free convolution.

Introduction Coxeter groups and noncrossing partitions $k$-divisible noncrossing partitions The classical types Fuss-Catalan combinatorics Bibliography.

Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

Probably the most prominent among the special integers that arise in combinatorial contexts are the binomial coefficients (~). These have many uses and, often, fascinating interpretations [9]. We would like to stress one particular interpretation in terms of paths on the integral lattice in the coordinate plane, and discuss the celebrated ballot problem using this interpretation. A path is a sequence of points Po P1 9 9 9 Pro, m >I O, where each P, is a lattice point (that is, a point with integer coordinates) and Pz+l, i 1> 0, is obtained by stepping one unit east or one unit north of P,. We say that this is a path from P to Q if Po = P, Pm= Q. It is now easy to count the number of paths.

http://math.uakron.edu/~cossey/636papers/hilton%20and%20pedersen.pdf

Catalan Numbers, Their Generalization, and Their Uses

Consider the following number trick–try it out on your friends. You ask them to write down the numbers from 0 to 9. Against 0 and 1 they write any two numbers (we suggest two fairly small positive integers just to avoid tedious arithmetic, but all participants should write the same pair of numbers). Then against 2 they write the sum of the entries against 0 and 1; against 3 they write the sum of the entries against 1 and 2; and so on. Once they have completed the process, producing entries against each number from 0 to 9, you suggest that, as a check, they call out the entry against the number 6. Thus their table (which, of course, you do not see) might look like the table in the margin. You now ask them to add all the entries in the second column, while you write 341 quickly on a slip of paper.

Fibonacci and Lucas Numbers

Part I. Personal Reflections On Investigation, Discovery, and Proof: 1. Discovery and dissection of a geometric gem Douglas R. Hofstadter 2. The role and function of proof in dynamic geometry: some personal reflections Michael de Villiers 3. Dynamic geometry renews interest in an old problem Dan Bennett 4. Dynamic geometry as a bridge from Euclidean geometry to analysis Albert A. Cuoco and E. Paul Goldenberg Part II. Dynamic Geometry In The Classroom: 5. Dynamic visualization from middle school through College James Morrow 6. Geometer's sketchpad in the classroom Tim Garry 7. Students discovering geometry using dynamic geometry software Michael Keyton 8. Moving triangles 9. Experiences with geometer's sketchpad in the classroom Kathryn W. Boehm 10. Beyond elementary constructions: selected exercises Arnold Perham and Bernadette H. Perham 11. Interactive generation, manipulation, and application of loci Heinz Schumann 12. Calculus with dynamic geometry Catherine A. Gorini 13. Beginning geometry at college Tony Hampson 14. Identifying transformations by their orbits James M. Parks 15. Dynamic proofs that use similarities James King 16. Visualization of group theory concepts through dynamic geometry Doris Schattschneider 17. Using the geometer's sketchpad with preservice teachers Zhonghong Jian and Edwin McClintock 18. Fish in the pond: inquiry with dynamic geometry Fadia Harik Part III. Dynamic Visualization In History, Perception, Optics and Aerodynamics: 19. Drawing logarithmic curves with geometer's sketchpad: a method inspired by historical sources David Dennis and Jere Confrey 20. Lost in the funhouse: an application of dynamic projective geometry Susan Addington and Stuart Levy 21. The use of dynamic geometry software in teaching and research in optometry and vision science Benjamin T. Backus 22. Creating airfoils from circles: the Joukowski transformation John Olive Part IV. The Worlds Of Dynamic Geometry: Issues In Design And Use: 23. Drawing worlds: scripted exploration environments in the geometer's sketchpad R. Nicholas Jackiw 24. Dynamic geometry and declarative geometric programming Richard J. Allen and Laurent Trilling.

Geometry turned on! : dynamic software in learning, teaching, and research

Fear No More

1 Going Down the Drain.- 1.1 Constructions.- 1.2 Cobwebs.- 1.3 Consolidation.- 1.4 Fibonacci Strikes.- 1.5 Denouement.- Final Break.- References.- Answers for Final Break.- 2 A Far Nicer Arithmetic.- 2.1 General Background: What You Already Know.- 2.2 Some Special Moduli: Getting Ready for the Fun.- 2.3 Arithmetic mod p: Some Beautiful Mathematics.- 2.4 Arithmetic mod Non-primes: The Same But Different.- 2.5 Primes, Codes, and Security.- 2.6 Casting Out 9's and 11's: Tricks of the Trade.- Final Break.- Answers for Final Break.- 3 Fibonacci and Lucas Numbers.- 3.1 A Number Trick.- 3.2 The Explanation Begins.- 3.3 Divisibility Properties.- 3.4 The Number Trick Finally Explained.- 3.5 More About Divisibility.- 3.6 A Little Geometry!.- Final Break.- References.- Answers for Final Break.- 4 Paper-Folding and Number Theory.- 4.1 Introduction: What You Can Do With-and Without-Euclidean Tools.- I Simple Paper-Folding.- 4.2 Going Beyond Euclid: Folding 2-Period Regular Polygons.- 4.3 Folding Numbers.- 4.4 Some Mathematical Tidbits.- II General Paper-Folding.- 4.5 General Folding Procedures.- 4.6 The Quasi-Order Theorem.- 4.7 Appendix: A Little Solid Geometry.- Final Break.- References.- 5 Quilts and Other Real-World Decorative Geometry.- 5.1 Quilts.- 5.2 Variations.- 5.3 Round and Round.- 5.4 Up the Wall.- Final Break.- References.- Answers for Final Break.- 6 Pascal, Euler, Triangles, Windmills.- 6.1 Introduction: A Chance to Experiment.- I Pascals Set the Scene.- 6.2 The Binomial Theorem.- 6.3 The Pascal Triangle and Windmill.- 6.4 The Pascal Flower and the Generalized Star of David.- II Euler Takes the Stage.- 6.5 Eulerian Numbers and Weighted Sums.- 6.6 Even Deeper Mysteries.- References.- 7 Hair and Beyond.- 7.1 A Problem with Pigeons, and Related Ideas.- 7.2 The Biggest Number.- 7.3 The Big Infinity.- 7.4 Other Sets of Cardinality ?0.- 7.5 Schroder and Bernstein.- 7.6 Cardinal Arithmetic.- 7.7 Even More Infinities?.- Final Break.- References.- Answers for Final Break.- 8 An Introduction to the Mathematics of Fractal Geometry.- 8.1 Introduction to the Introduction: What's Different About Our Approach.- 8.2 Intuitive Notion of Self-Similarity.- 8.3 The lent Map and the Logistic Map.- 8.4 Some More Sophisticated Material.- Final Break.- References.- Answers for Final Break.- An Introduction to the Mathematics of Fractal Geometry.- 8.1 Introduction to the Introduction: What's Different About Our Approach.- 8.2 Intuitive Notion of Self-Similarity.- 8.3 The tent Map and and the Logistic Map.- 8.4 Some more Sophisticated Material.- Final Break.- References.- Answer for Final Break.- 9 Some of Our Own Reflections.- 9.1 General Principles.- 9.2 Specific Principles.- 9.3 Appendix: Principles of Mathematical Pedagogy.- References.

Jean Pedersen

Papers

Catalan Numbers, Their Generalization, and Their Uses

Fibonacci and Lucas Numbers

Geometry turned on! : dynamic software in learning, teaching, and research

Fear No More

Mathematical Reflections: In a Room with Many Mirrors