Classical and Umbral Moonshine: Connections and pp-adic Properties

The classical theory of monstrous moonshine describes the unexpected connection between the representation theory of the monster group $M$, the largest of the simple sporadic groups, and certain modular functions, called Hauptmodln. In particular, the $n$-th Fourier coefficient of Klein's $j$-function is the dimension of the grade $n$ part of a special infinite dimensional representation $V$ of the monster group. More generally the coefficients of Hauptmoduln are graded traces $T_g$ of $g \in M$ acting on $V$. Similar phenomena have been shown to hold for the Matthieu group $M_{24}$, but instead of modular functions, mock modular forms must be used. This has been conjecturally generalized even further, to umbral moonshine, which associates to each of the 23 Niemeier lattices a finite group, infinite dimensional representation, and mock modular form. We use generalized Borcherds products to relate monstrous moonshine and umbral moonshine. Namely, we use mock modular forms from umbral moonshine to construct via generalized Borcherds products rational functions of the Hauptmoduln $T_g$ from monstrous moonshine. This allows us to associate to each pure $A$-type Niemeier lattice a conjugacy class $g$ of the monster group, and gives rise to identities relating dimensions of representations from umbral moonshine to values of $T_g$. We also show that the logarithmic derivatives of the Borcherds products are $p$-adic modular forms for certain primes $p$ and describe some of the resulting properties of their coefficients modulo $p$.Comment: 21 pages, to appear in the Journal of the Ramanujan Mathematical Societ

Hermite-interpolatory subdivision schemes

Stationary interpolatory subdivision schemes for Hermite data that consist of function values and first derivatives are examined. A general class of Hermite-interpolatory subdivision schemes is proposed, and some of its basic properties are stated. The goal is to characterise and construct certain classes of nonlinear (and linear) Hermite schemes. For linear Hermite subdivision, smoothness conditions known from the literature are discussed. In order to allow a simpler construction of suitable nonlinear Hermite subdivision schemes, these conditions are posed as assumptions. For linear Hermite subdivision, explicit schemes that satisfy sufficient conditions for $C^2$-convergence are constructed. This leads to larger classes of $C^2$ schemes than known from the literature. Finally, convexity preserving Hermite-interpolatory subdivision is examined, and some explicit rational schemes that generate $C^1$ limit functions are presented

Dynamics in the Eremenko-Lyubich class

The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials -- in other words transcendental entire functions -- is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics. In this paper we survey the dynamics of functions in the Eremenko-Lyubich class, $\mathcal{B}$. Among transcendental entire functions, those in this class have properties that make their dynamics most "similar" to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of class $\mathcal{B}$ functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases. We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. For any (1+1)-dimensional scalar evolution equation $E$, we define a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations. As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a B\"acklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for B\"acklund transformations are presented in other publications as well. In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs. In this paper we describe general properties of $F(E)$ and present generators and relations for these algebras. In other publications we study the structure of $F(E)$ for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.Comment: 23 pages; v4: some results have been moved to other preprint

Newton flows for elliptic functions

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e., doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions $f$ of fixed order $r$ ($\geq$ 2) we prove: For almost all functions $f$, the corresponding Newton flows are structurally stable i.e., topologically invariant under small perturbations of the zeros and poles for $f$ [genericity]. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded, graph $G(f)$ on $T$ with $r$ vertices, 2$r$ edges and $r$ faces that fulfil certain combinatorial properties (Euler, Hall) on some of its subgraphs. The graph $G(f)$ determines the conjugacy class of the flow [characterization]. A connected, cellularly embedded toroidal graph $G$ with the above Euler and Hall properties, is called a Newton graph. Any Newton graph $G$ can be realized as the graph $G(f)$ of the structurally stable Newton flow for some function $f$ [classification]. This leads to: up till conjugacy between flows and(topological) equivalency between graphs, there is a 1-1 correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order $r$ of the underlying functions $f$ [representation]. In particular, it follows that in case $r$ = 2, there is only one (up to conjugacy) structurally stabe elliptic Newton flow, whereas in case $r$ = 3, we find a list of nine graphs, determining all possibilities. Moreover, we pay attention to the so-called nuclear Newton flows of order $r$, and indicate how - by a bifurcation procedure - any structurally stable elliptic Newton flow of order $r$ can be obtained from such a nuclear flow. Finally, we show that the detection of elliptic Newton flows is possible in polynomial time. The proofs of the above results rely on Peixoto's characterization/classication theorems for structurally stable dynamical systems on compact 2-dimensional manifolds, Stiemke's theorem of the alternatives, Hall's theorem of distinct representatives, the Heter-Edmonds-Ringer rotation principle for embedded graphs, an existence theorem on gradient dynamical systems by Smale, and an interpretation of Newton flows as steady streams

Applications of continued fractions in one and more variables

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Elementary properties of continued fractions are derived from sets of three-term recurrence relations and approximation methods are developed from this simple approach. First, a well-known method for numerical inversion of Laplace transforms is modified in two different ways to obtain exponential approximations. Differential-difference equations arising from certain Markov processes are solved by direct application of continued fractions and practical error estimates are obtained. Approximations of a slightly different form are then derived for certain generalised hypergeometric functions using those hypergeometric functions that satisfy three-term recurrence relations and have simple continued fraction expansions. Error estimates are also given in this case. The class of corresponding sequence algorithms is then described for the transformation of power series into continued fraction form. These algorithms are shown to have very general application and only break down if the required continued fraction does not exist. A continued fraction in two variables is then shown to exist and its correspondence with suitable double power series made feasible by the generalisation of the corresponding sequence method. A convergence theorem, due to Van Vleck, is adapted for use with this type of continued fraction and a comparison is made with Chisholm rational approximants in two variables. Some of these ideas are further generalised to the multivariate case. Such corresponding fractions are closely related to other fractions that may be used for point-wise bivariate or multivariate interpolation to function values known on a mesh of points. Interpolation algorithms are described and advantages and limitations discussed. The work presented forms a basis for a wide range of further research and some possible applications in numerical mathematics are indicated

Newton flows for elliptic functions II:Structural stability: classification and representation

In our previous paper we associated to each non-constant elliptic function f on a torus T a dynamical system, the elliptic Newton flow corresponding to f. We characterized the functions for which these flows are structurally stable and showed a genericity result. In the present paper we focus on the classification and representation of these structurally stable flows. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded graph G(f) on a torus T with r vertices, 2r edges and r faces that fulfil certain combinatorial properties (Euler, Hall) on some of its subgraphs. The graph G(f) determines the conjugacy class of the flow [classification]. A connected, cellularly embedded toroidal graph G with the above Euler and Hall properties, is called a Newton graph. Any Newton graph G can be realized as the graph G(f) of the structurally stable Newton flow for some function f. This leads to: up till conjugacy between flows and (topological) equivalency between graphs, there is a one to one correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order r of the underlying functions f [representation]. Finally, we clarify the analogy between rational and elliptic Newton flows, and show that the detection of elliptic Newton flows is possible in polynomial time. The proofs of the above results rely on Peixoto’s characterization/classification theorems for structurally stable dynamical systems on compact 2-dimensional manifolds, Stiemke’s theorem of the alternatives, Hall’s theorem of distinct representatives, the Heffter–Edmonds–Ringer rotation principle for embedded graphs, an existence theorem on gradient dynamical systems by Smale, and an interpretation of Newton flows as steady streams