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

High-resolution palynological analysis in Lake Sapanca as a tool to detect recent earthquakes on the North Anatolian Fault

High-resolution palynological analysis of a 38-cm-long core collected from Lake Sapanca, northwest Turkey, reveals large earthquakes that occurred during the second half of the 20th century along the North Anatolian Fault Zone. Four events have disturbed the lacustrine sedimentary sequence. Three of the four events are historical earthquakes in 1999 in Izmit, 1967 in Mudurnu and 1957 in Abant. These events are recorded in the core by turbiditic deposits and reworked sediment and by low overall palynomorph concentrations but high values of thick-exined pollen, fern spores and fungal spores. Palynomorphs in the event beds have been grouped based on their associations in modern moss, river and lake samples. The inferred mechanisms of transport and sources for the palynomorphs are: 1- lake sediment displaced by slump, 2- collapsed shoreline sediment owing to seiche, waves and sudden lake level changes, 3- subsidence of deltas and 4- river-transported soil and sediment from upland areas. The 1999 Izmit earthquake is only weakly recorded by palynomorphs, probably due to recent engineering control on the rivers. The 1967 Mudurnu earthquake had the strongest effect on the lake, introducing successive packages of sediment to the centre of the lake from underwater slopes, the lakeshore and rivers

Economic Activity Associated With the Garrison Diversion Unit in 1984

Resource /Energy Economics and Policy,