Noncommutative symmetric functions and W-polynomials

Let K,S,D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Vieta formula and decompositions of differential operators. W-polynomials show up naturally, their connections with P-independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations of W-polynomials are analysed. Connections between the existence of LLCM of monic linear polynomials with coefficients in a ring and the left duo property are established at the end of the paper

Complete hyperfine Paschen-Back regime at relatively small magnetic fields realized in Potassium nano-cell

A one-dimensional nano-metric-thin cell (NC) filled with potassium metal has been built and used to study optical atomic transitions in external magnetic fields. These studies benefit from the remarkable features of the NC allowing one to use $\lambda/2$- and $\lambda$-methods for effective investigations of individual transitions of the K D_1 line. The methods are based on strong narrowing of the absorption spectrum of the atomic column of thickness L equal to $\lambda/2$ and to $\lambda$(with \lambda = 770\un{nm} being the resonant laser radiation wavelength). In particular, for a $\pi$-polarized radiation excitation the $\lambda$-method allows us to resolve eight atomic transitions (in two groups of four atomic transitions) and to reveal two remarkable transitions that we call Guiding Transitions (GT). The probabilities of all other transitions inside the group (as well as the frequency slope versus magnetic field) tend to the probability and to the slope of GT. Note that for circular polarization there is one group of four transitions and GT do not exist. Among eight transitions there are also two transitions (forbidden for $B$ = 0) with the probabilities undergoing strong modification under the influence of magnetic fields. Practically the complete hyperfine Paschen-Back regime is observed at relatively low (\sim 1\un{kG}) magnetic fields. Note that for K $D_2$ line GT are absent. Theoretical models describe the experiment very well.Comment: 6 page

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presente

The uses of qualitative data in multimethodology:Developing causal loop diagrams during the coding process

In this research note we describe a method for exploring the creation of causal loop diagrams (CLDs) from the coding trees developed through a grounded theory approach and using computer aided qualitative data analysis software (CAQDAS). The theoretical background to the approach is multimethodology, in line with Minger’s description of paradigm crossing and is appropriately situated within the Appreciate and Analyse phases of PSM intervention. The practical use of this method has been explored and three case studies are presented from the domains of organisational change and entrepreneurial studies. The value of this method is twofold; (i) it has the potential to improve dynamic sensibility in the process of qualitative data analysis, and (ii) it can provide a more rigorous approach to developing CLDs in the formation stage of system dynamics modelling. We propose that the further development of this method requires its implementation within CAQDAS packages so that CLD creation, as a precursor to full system dynamics modelling, is contemporaneous with coding and consistent with a bridging strategy of paradigm crossing