486 research outputs found
On transcendental numbers: new results and a little history
Attempting to create a general framework for studying new results on
transcendental numbers, this paper begins with a survey on transcendental
numbers and transcendence, it then presents several properties of the
transcendental numbers and , and then it gives the proofs of new
inequalities and identities for transcendental numbers. Also, in relationship
with these topics, we study some implications for the theory of the Yang-Baxter
equations, and we propose some open problems.Comment: 8 page
Minimal rational curves on wonderful group compactifications
Consider a simple algebraic group G of adjoint type, and its wonderful
compactification X. We show that X admits a unique family of minimal rational
curves, and we explicitly describe the subfamily consisting of curves through a
general point. As an application, we show that X has the target rigidity
property when G is not of type A_1 or C.Comment: 16
Fourier transforms of Gibbs measures for the Gauss map
We investigate under which conditions a given invariant measure for the
dynamical system defined by the Gauss map is a Rajchman
measure with polynomially decaying Fourier transform We show that this
property holds for any Gibbs measure of Hausdorff dimension greater than
with a natural large deviation assumption on the Gibbs potential. In
particular, we obtain the result for the Hausdorff measure and all Gibbs
measures of dimension greater than on badly approximable numbers, which
extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result
implies that the Fourier-Stieltjes coefficients of the Minkowski's question
mark function decay to polynomially answering a question of Salem from
1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain
an equidistribution theorem for Gibbs measures, which extends in part a recent
result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear
and number theoretic nature of the Gauss map and large deviation theory for
Hausdorff dimension and Lyapunov exponents.Comment: v3: 29 pages; peer-reviewed version, fixes typos and added more
elaborations, and included comments on Salem's problem. To appear in Math.
An
The Theory of Quasiconformal Mappings in Higher Dimensions, I
We present a survey of the many and various elements of the modern
higher-dimensional theory of quasiconformal mappings and their wide and varied
application. It is unified (and limited) by the theme of the author's
interests. Thus we will discuss the basic theory as it developed in the 1960s
in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore
the connections with geometric function theory, nonlinear partial differential
equations, differential and geometric topology and dynamics as they ensued over
the following decades. We give few proofs as we try to outline the major
results of the area and current research themes. We do not strive to present
these results in maximal generality, as to achieve this considerable technical
knowledge would be necessary of the reader. We have tried to give a feel of
where the area is, what are the central ideas and problems and where are the
major current interactions with researchers in other areas. We have also added
a bit of history here and there. We have not been able to cover the many recent
advances generalising the theory to mappings of finite distortion and to
degenerate elliptic Beltrami systems which connects the theory closely with the
calculus of variations and nonlinear elasticity, nonlinear Hodge theory and
related areas, although the reader may see shadows of this aspect in parts
Population Dynamics and Non-Hermitian Localization
We review localization with non-Hermitian time evolution as applied to simple
models of population biology with spatially varying growth profiles and
convection. Convection leads to a constant imaginary vector potential in the
Schroedinger-like operator which appears in linearized growth models. We
illustrate the basic ideas by reviewing how convection affects the evolution of
a population influenced by a simple square well growth profile. Results from
discrete lattice growth models in both one and two dimensions are presented. A
set of similarity transformations which lead to exact results for the spectrum
and winding numbers of eigenfunctions for random growth rates in one dimension
is described in detail. We discuss the influence of boundary conditions, and
argue that periodic boundary conditions lead to results which are in fact
typical of a broad class of growth problems with convection.Comment: 19 pages, 11 figure
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