This work aims to prove a desingularization theorem for analytic families of two-dimensional vector fields, under the hypothesis that all its singularities have a non-vanishing first jet. Application to problems of singular perturbations and finite cyclicity are discussed in the last chapter.

Starting from Borcherds' fake monster Lie algebra, this text construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including $AE_3$.

The author explores ramifications and extensions of a $q$-difference operator method first used by L.J. Rogers for deriving relationships between special functions involving certain fundamental $q$-symmetric polynomials. In special cases these symmetric polynomials reduce to well-known classes of orthogonal polynomials. A number of basic properties of these polynomials follow from this approach. This leads naturally to the evaluation of the Askey-Wilson integral and generalizations. Expansions of certain generalized basic hypergeometric functions in terms of the symmetric polynomials are also...

This book is intended for graduate students and research mathematicians interested in dynamical systems and ergodic theory.

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One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in $ZFC$. The authors' starting point is the following elementary, though non-trivial result: Consider $X subset 2omega imes2omega$, set $Y=pi(X)$, where $pi$ denotes the canonical projection of $2omega imes2omega$ onto the first factor, and suppose that $(star)$: Any compact subset of $Y$ is the projection of some compact subset of $X$. If moreover $X$ is $mathbf{Pi 0 2$ then $(starstar)$: The restriction of $pi$ to some relatively closed subset of $X$ is perfect onto $Y$ it follows...

Proves Rivoal's 'denominator conjecture' concerning the common denominators of coefficients of certain linear forms in zeta values; these forms were constructed to obtain lower bounds for the dimension of the vector space over $mathbb Q$ spanned by $1,zeta

Shows that for finite systems of quadratic exponential equations decidability passes, under certain hypotheses, from the factor groups to free products and one-relator products.

Focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form its semi group, the gradient of the semi group, functional calculus, square functions and Riesz transforms. This book introduces four numbers associated to the semi

Presents categorical foundations that are needed for working out completely the rigorous approach to the definition of conformal field theory outlined by Graeme Segal. This book discusses pseudo algebras over theories and 2-theories, their pseudo morphisms