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This chapter surveys recent progress on the enumeration of labelled planar graphs and on the distribution of several parameters in random planar graphs. It shows how to use generating functions to obtain asymptotic estimates and limit probability laws. Emphasis is made on presenting the main ideas while keeping technical details to a minimum.

The previous chapter looked at the classic random graph model, in which pairs of vertices are connected at random with uniform probabilities. Although this model has proved tremendously useful as a source of insight into the structure of networks, it also has a number of serious shortcomings. Chief among these is its degree distribution, which follows the Poisson distribution which is quite different from the degree distributions seen in most real-world networks. This chapter shows how to create more sophisticated random graph models, which incorporate arbitrary degree distributions and yet are still exactly solvable for many of their properties in the limit of large network size. The fundamental mathematical tool used to derive the results of this chapter is the probability generating function. Exercises are provided at the end of the chapter.

In the previous chapter, several conditions were given that could be used for testing transformations to see whether they were canonical or not. But no methods, other than trial and error, were given for actually creating the canonical transformations to be tested. In this chapter, methods for creating transformations that will automatically be canonical are presented. Canonical transformations can be created by first choosing what are called generating functions. Using one of these generating functions in the formalism to be described will generate a transformation that will be canonical by construction. The generating functions can be quite general, leading to a wide selection of possible canonical transformations. This chapter also discusses proto-generating functions, examples of generating functions, mixed generating functions, simple generating functions, traditional generating functions, standard form of extended Hamiltonian recovered, differential canonical transformations, active canonical transformations, phase-space analog of Noether theorem, and Liouville theorem.

In 1857, while investigating the partition number function, J. J. Sylvester defined the function d(m; a1, . . . , an), called the denumerant, as the number of nonnegative integer representations of m by a1, . . . , an. This chapter is devoted to the study of the denumerant and related functions. After discussing briefly some basic properties of the partition function and its relation with denumerants, the general behaviour of d(m; a1, . . . , an) and its connection to g(a1, . . . , an) are analyzed. Two interesting methods for computing denumerants — one based on a decomposition of the rational fraction into partial fractions and another due to E. T. Bell — are described. An exact value of d(m; p, q) — first found by T. Popoviciu in 1953 — is proved, and the known results when n = 2 and n = 3 are summarized. The calculation of g(a1, . . . , an) by using Hilbert series via free resolutions, and the use of this approach to show an explicit formula for g(a1, a2, a3), are shown. The connection among denumerants, FP, and Ehrhart polynomial as well as two variants of d(m; a1, . . . , an) are discussed.

This chapter presents important results obtained for the Casimir effect in the presence of spherical and cylindrical shells with various boundary conditions. It also includes the Casimir effect for a dielectric ball. The configuration of a spherical shell finds applications in the bag model of quantum chromodynamics. Mode summation for both interior and exterior regions is considered. Analytic continuation for obtaining the regularized vacuum energy, and the divergent contribution are analyzed. This allows for the renormalized vacuum energy for scalar, Electromagnetic, and spinor fields to be found. The case of nonzero temperature is considered on the basis of the general formalism presented in Chapter 5. The material in this chapter is focused on the technical methods necessary to calculate the Casimir effect in spherical and cylindrical geometry and on the analysis of the ultraviolet divergences.

After James Bernoulli, the main contributors to probability theory and its applications in the 18th century worked either to solve more intricate problems of games of chance (De Moivre), or to build new probability models (De Moivre, Daniel Bernoulli). Further, some of them used probabilistic arguments to test hypotheses about or to estimate parameters involved in probability models for real life phenomena in the fields of demography, astronomy, and theory of errors. The concept of continuous variables, the tool of generating functions, the normal model approximation to the binomial, and ‘the rational expectation principle’ (in the context of the St. Petersburg Paradox) emerged out of these studies.

This chapter deals with BLAST theory. BLAST (Basic Local Alignment Search Tool) is a widely used statistical method for finding similarities between sequences of symbols from finite alphabets. While the theory is completely general, the most widely used applications are to comparing sequences of nucleotides and sequences of amino acids. The fundamental objective of BLAST theory is to align sequences as well as possible, and then make a determination as to the level of statistical significance of the alignment. Thus one computes a “maximal segmental score” of the alignment between the two sequences, and tests to see whether the maximal segmental score could have been obtained purely as a matter of chance. The chapter presents the main results of BLAST theory, focusing on the moment generating function and application of the results. It also presents the proofs of the main results.

This chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in the last section of the chapter. The moment generating function is developed for studying the photon number statistics of a single field mode. The quantum properties of optical phase are described using the optical phase operator. The characteristic functions and quasi-probability distributions provide a complete statistical description of the field. These rely on the properties the coherent states and the Glauber displacement operator.

This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a p-adic group can be replaced by a sum over Kashiwara's crystal. Partly motivated by the crystal description presented in Chapter 2 of this book, this perspective was advocated by Bump and Nakasuji. Later work by McNamara and Kim and Lee extended this philosophy yet further. Indeed, McNamara shows that the computation of the metaplectic Whittaker function is initially given as a sum over Kashiwara's crystal. The chapter considers Kostant's generating function, the character of the quantized enveloping algebra, and its association with Kashiwara's crystal, along with the Kostant partition function and the Weyl character formula.

This chapter introduces, in the case of ordinary integrals, concepts and methods that can be generalized to path integrals. The first part is devoted to the calculation of ordinary Gaussian integrals, Gaussian expectation values, and the proof of the corresponding Wick's theorem. The notion of connected contributions is discussed, and it is shown that expectation values of monomials can conveniently be associated with graphs called Feynman diagrams. Expectation values corresponding to measures that deviate slightly from Gaussian measures can be reduced to sums of infinite series of Gaussian expectation values, a method known as perturbation theory. This chapter also contains a short presentation of the steepest descent method, a method that allows evaluating a class of integrals by reducing them, in some limit, to Gaussian integrals. To discuss properties of expectation values with respect to some measure or probability distribution, it is always convenient to introduce a generating function of the moments of the distribution.

This chapter presents a number of technical results concerning generating functions, Gaussian measures, and the steepest descent method. It begins by introducing the notion of generating function of the moments of a probability distribution. It then calculates Gaussian integrals and prove Wick′s theorem for Gaussian expectation values, a result that is simple but of major practical importance. The steepest descent method provides asymptotic evaluations, in some limits, of real or complex integrals. It leads to calculations of Gaussian expectation values, which explains its presence in this chapter. Moreover, the steepest descent method will be directly useful in this work. Exercises are provided at the end of the chapter.

This chapter reviews probability theory, limiting the discussion to the study of random events as opposed to random processes, the latter being a sequence of random events extended over a period of time. The goal is to raise the level of approach by demonstrating the usefulness of delta functions. The chapter presents a calculation of the chi-squared distribution (important in statistical decision making) with delta functions. The normalisation condition of the probability density in chi-square leads to a geometric result; namely, the volume of a sphere in n dimensions can be determined without ever transferring to spherical coordinates. This chapter also discusses the first and second laws of gambling, along with distribution functions, stochastic variables, expectation values for single random variables, characteristic functions and generating functions, measures of dispersion, joint events, conditional probabilities and Bayes' theorem, sums of random variables, fitting of experimental observations, and multivariate normal distributions.

This chapter proposes methods for creating transformations that will automatically be canonical. Canonical transformations can be created by first choosing what are called generating functions. Using one of these generating functions in the formalism to be described will generate a transformation that will be canonical by construction. The generating functions can be quite general, leading to a wide selection of possible canonical transformations. Not only does every generating function lead to a canonical transformation, the converse is also true. Given any canonical transformation, a generating function can always be found that will generate it. The chapter begins by first relating canonical transformations to the existence of an intermediate function that will be called a protogenerating function.

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

This chapter concentrates on the properties of the walks which do not immediately follow from the distribution of the particle's position after n steps, like the first passage and return probabilities. These are most elegantly obtained via generating functions of the corresponding distributions. Again, the necessary mathematical toolbox is first introduced and discussed, including definition and main properties of generating functions and Tauberian theorems giving a nice method of approximate restoration of probability distributions from their generating functions. Then it concentrates on explicit calculations of first passage and return probabilities of the lattice walks, as well on the mean number of distinct sites visited by a walk of n steps. Some properties of off‐lattice walks are discussed as well.

This chapter discusses generating functions in more detail. It shows how generating functions give rise to discrete-time analogues of the symplectic action functional and hence lead to discrete variational problems. The results of this chapter form the basis for the proofs in Chapter 11 of the Arnold conjecture for the torus and in Chapter 12 of the existence of the Hofer–Zehnder capacity. The final section examines generating functions for exact Lagrangian submanifolds of cotangent bundles.

Special functions are advanced functions frequently used in science and engineering. The properties of Legendre, Hermite, Laguerre, Chebyshev and spherical harmonics polynomials and Bessel functions are obtained. The use of generating functions, recursion formulas, differential relations, Rodrigues formulas and differential equations to define and manipulate these functions are described. Their general treatments as classical orthogonal polynomials and as Sturm Liouville eigenfunctions are given.

Interactions between the environment, traits, and their constituent genes demonstrate that it is impossible to predict evolutionary change in anything but the simplest environments. Even then, we can predict the expected course of evolution only by mapping a trait’s values onto fitness (the fitness-mapping function). If individuals exploit more than a single environment (or habitat), then we can use the set of fitness values in each one to help explore whether evolution will favour specialized or generalized phenotypes. The solution requires more than the fitness-mapping function. We need to also generate an adaptive function that reveals the average adaptive value of each trait in the mix of environments that it exploits. One of the interesting lessons that can be learnt from this approach is that density changes the fitness-mapping function, and frequency alters its adaptive value.