Approximation by Polynomials with Integral Coefficients

Results in the approximation of functions by polynomials with coefficients which are integers have been appearing since that of Pal in 1914. The body of results has grown to an extent which seems to justify this book. The intention here is to make these results as accessible as possible. The book addresses essentially two questions. The first is the question of what functions can be approximated by polynomials whose coefficients are ...

Character Identities in the Twisted Endoscopy of Real Reductive Groups

Suppose G is a real reductive algebraic group, ? is an automorphism of G, and ? is a quasicharacter of the group of real points G(R). Under some additional assumptions, the theory of twisted endoscopy associates to this triple real reductive groups H. The Local Langlands Correspondence partitions the admissible representations of H(R) and G(R) into L-packets. The author proves twisted character identities between L-packets of H(R) an...

On the Shape of a Pure O-sequence

A monomial order ideal is a finite collection $X$ of (monic) monomials such that, whenever $M\in X$ and $N$ divides $M$, then $N\in X$. Hence $X$ is a poset, where the partial order is given by divisibility. If all, say $t$, maximal monomials of $X$ have the same degree, then $X$ is pure (of type $t$). A pure $O$-sequence is the vector, $\underline{h}=(h_0=1,h_1,...,h_e)$, counting the monomials of $X$ in each degree. Equivalently, p...

Stochastic Partial Differential Equations: Six Perspectives

The field of Stochastic Partial Differential Equations (SPDEs) is one of the most dynamically developing areas of mathematics. It lies at the cross section of probability, partial differential equations, population biology, and mathematical physics. The field is especially attractive because of its interdisciplinary nature and the enormous richness of current and potential future applications. This volume is a collection of six impor...

Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration

This book deals first with Haar bases, Faber bases and Faber frames for weighted function spaces on the real line and the plane. It extends results in the author's book, "Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration" (EMS, 2010), from unweighted spaces (preferably in cubes) to weighted spaces. The obtained assertions are used to study sampling and numerical integration in weighted spaces on the real line and...

Vector Bundles on Degenerations of Elliptic Curves and Yang-Baxter Equations

In this paper the authors introduce the notion of a geometric associative $r$-matrix attached to a genus one fibration with a section and irreducible fibres. It allows them to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. They also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bu...

Non-Cooperative Equilibria of Fermi Systems with Long Range Interactions

The authors define a Banach space $\mathcal{M}_{1}$ of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalised) equilibrium states for any $\mathfrak{m}\in \mathcal{M}_{1}$. In particular, the authors give a first answer to an old open problem in mathematical physics--first addressed by Ginibre in 1968 within a different context - about the validity of the so-ca...

The Reflective Lorentzian Lattices of Rank 3

The author classifies all the symmetric integer bilinear forms of signature $(2,1)$ whose isometry groups are generated up to finite index by reflections. There are 8,595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin's enumeration of the strongly square-free cases. The author's technique is an analysis of the shape of the Weyl chamber, followed by computer work using V...

Seifert Fiberings

Seifert fiberings extend the notion of fiber bundle mappings by allowing some of the fibers to be singular. Away from the singular fibers, the fibering is an ordinary bundle with fiber a fixed homogeneous space. The singular fibers are quotients of this homogeneous space by distinguished groups of homeomorphisms. These fiberings are ubiquitous and important in mathematics. This book describes in a unified way their structure, how the...