The Stokes Phenomenon and Some Applications

Multisummation provides a transparent description of Stokes matrices which is reviewed here together with some applications. Examples of moduli spaces for Stokes matrices are computed and discussed. A moduli space for a third Painlev\'e equation is made explicit. It is shown that the monodromy identity, relating the topological monodromy and Stokes matrices, is useful for some quantum differential equations and for confluent generalized hypergeometric equations

A theory of structural model validity in simulation.

During the last decennia, the practice of simulation has become increasingly popular among many system analysts, model builders and general scientists for the purpose of studying complex systems that surpass the operability of analytical solution techniques. As a consequence of the pragmatic orientation of simulation, a vital stage for a successful application is the issue of validating a constructed simulation model. Employing the model as an effective instrument for assessing the benefit of structural changes or for predicting future observations makes validation an essential part of any productive simulation study. The diversity of the employment field of simulation however brings about that there exists an irrefutable level of ambiguity concerning the principal subject of this validation process. Further, the literature has come up with a plethora of ad hoc validation techniques that have mostly been inherited from standard statistical analysis. It lies within the aim of this paper to reflect on the issue of validation in simulation and to present the reader with a topological parallelism of the classical philosophical polarity of objectivism versus relativism. First, we will position validation in relation to verification and accreditation and elaborate on the prime actors in validation, i.e. a conceptual model, a formal model and behaviour. Next, we will formally derive a topological interpretation of structural validation for both objectivists and relativists. As will be seen, recent advances in the domain of fuzzy topology allow for a valuable metaphor of a relativistic attitude towards modelling and structural validation. Finally, we will discuss several general types of modelling errors that may occur and examine their repercussion on the natural topological spaces of objectivists and relativists. We end this paper with a formal, topological oriented definition of structural model validity for both objectivists and relativists. The paper is concluded with summarising the most important findings and giving a direction for future research.Model; Simulation; Theory; Scientists; Processes; Statistical analysis;

Galois theory of q-difference equations

Choose $q\in {\mathbb C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a functorial way a vector bundle v(M) on the Tate curve $E_q:={\mathbb C}^*/q^{\mathbb Z}$. As a corollary one rediscovers Atiyah's classification of the indecomposable vector bundles on the complex Tate curve. Linear q-difference equations are also studied in positive characteristic in order to derive Atiyah's results for elliptic curves for which the j-invariant is not algebraic over ${\mathbb F}_p$. A universal difference ring and a universal formal difference Galois group are introduced. Part of the difference Galois group has an interpretation as `Stokes matrices', the above moduli space is the algebraic tool to compute it. It is possible to provide the vector bundle v(M) on E_q, corresponding to a difference module M over K, with a connection $\nabla_M$. If M is regular singular, then $\nabla_M$ is essentially determined by the absense of singularities and `unit circle monodromy'. More precisely, the monodromy of the connection $(v(M),\nabla_M)$ coincides with the action of two topological generators of the universal regular singular difference Galois group. For irregular difference modules, $\nabla_M$ will have singularities and there are various Tannakian choices for $M\mapsto (v(M),\nabla_M)$. Explicit computations are difficult, especially for the case of non integer slopes.Comment: Corrected versio

Different sensitivities of two optical magnetometers realized in the same experimental arrangement

In this article, operation of optical magnetometers detecting static (DC) and oscillating (AC) magnetic fields is studied and comparison of the devices is performed. To facilitate the comparison, the analysis is carried out in the same experimental setup, exploiting nonlinear magneto-optical rotation. In such a system, a control over static-field magnitude or oscillating-field frequency provides detection of strength of the DC or AC fields. Polarization rotation is investigated for various light intensities and AC-field amplitudes, which allows to determine optimum sensitivity to both fields. With the results, we demonstrate that under optimal conditions the AC magnetometer is about ten times more sensitive than its DC counterpart, which originates from different response of the atoms to the fields. Bandwidth of the magnetometers is also analyzed, revealing its different dependence on the light power. Particularly, we demonstrate that bandwidth of the AC magnetometer can be significantly increased without strong deterioration of the magnetometer sensitivity. This behavior, combined with the ability to tune the resonance frequency of the AC magnetometer, provide means for ultra-sensitive measurements of the AC field in a broad but spectrally-limited range, where detrimental role of static-field instability is significantly reduced.Comment: 9 pages, 6 figure

Solution of the general dynamic equation along approximate fluid trajectories generated by the method of moments

We consider condensing flow with droplets that nucleate and grow, but do not slip with respect to the surrounding gas phase. To compute the local droplet size distribution, one could solve the general dynamic equation and the fluid dynamics equations simultaneously. To reduce the overall computational effort of this procedure by roughly an order of magnitude, we propose an alternative procedure, in which the general dynamic equation is initially replaced by moment equations complemented with a closure assumption. The key notion is that the flow field obtained from this so-called method of moments, i.e., solving the moment equations and the fluid dynamics equations simultaneously, approximately accommodates the thermodynamic effects of condensation. Instead of estimating the droplet size distribution from the obtained moments by making assumptions about its shape, we subsequently solve the exact general dynamic equation along a number of selected fluid trajectories, keeping the flow field fixed. This alternative procedure leads to fairly accurate size distribution estimates at low cost, and it eliminates the need for assumptions on the distribution shape. Furthermore, it leads to the exact size distribution whenever the closure of the moment equations is exact

Mumford curves and Mumford groups in positive characteristic

A Mumford group is a discontinuous subgroup $\Gamma$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $\Gamma$ is a curve of genus 0. As abstract group $\Gamma$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to our paper "Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and K.~Kontogeorgis.Comment: 62 page