The Principal Branch of the Lambert W Function

Funder: University of CambridgeAbstractThe LambertWfunction is the multi-valued inverse of the function$$E(z) = z \exp z$$E(z)=zexpz. Let$$\widetilde{W}$$W~be a branch ofWdefined and single-valued on a region$$\widetilde{D}$$D~. We show how to use the Taylor expansion of$$\widetilde{W}$$W~at a given point of$$\widetilde{D}$$D~to obtain an infinite series representation of$$\widetilde{W}$$W~throughout$$\widetilde{D}$$D~.</jats:p

The parabola theorem on continued fractions

Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Möbius transformations. This geometric approach allows us to relate the Stern–Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Möbius transformations acting on three-dimensional hyperbolic space. We also obtain a version of the parabola theorem in several dimensions

Random iteration of analytic maps

We consider analytic maps Fj: D → D of a domain D into itself and ask when does the sequence f1 ο⋯ο fn converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence {w1, w2,...} in D whose values are not taken by any f j in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.published_or_final_versio

On the structure of acyclic binary relations

We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain)

Characteristic Energy of the Coulomb Interactions and the Pileup of States

Tunneling data on $\mathrm{La_{1.28}Sr_{1.72}Mn_2O_7}$ crystals confirm Coulomb interaction effects through the $\sqrt{\mathrm{E}}$ dependence of the density of states. Importantly, the data and analysis at high energy, E, show a pileup of states: most of the states removed from near the Fermi level are found between ~40 and 130 meV, from which we infer the possibility of universal behavior. The agreement of our tunneling data with recent photoemission results further confirms our analysis.Comment: 4 pages, 4 figures, submitted to PR

Open questions in utility theory

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE)

The Uniformisation of the Equation zw=wzz^w=w^z

AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .</jats:p

Winding Numbers, Unwinding Numbers, and the Lambert W Function

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.</jats:p