Markov chains with replacement

The following modification of a general state space discrete-time Markov chain is considered: certain transitions are supposed “forbidden” and the chain evolves until there is such a transition. At this instant the value of the chain is “replaced” according to a given rule, and, starting from the new value, the chain evolves normally until there is a forbidden transition again; the cycle is then repeated. The relationship of this modified process to the original one is studied in general terms, with particular emphasis being given to invariant measures. Examples are given which illustrate the results obtained.     

Genetic Algorithm Applied to the Selection of Conditions for the Simultaneous Quantification of Three-Food Colorants Using a Hand Scanner

A low-cost and reliable method employing a hand scanner for simultaneous colorimetric quantification of food colorant mixtures including amaranth, brilliant blue, and tartrazine is presented. Compared to a spectrophotometer, a hand scanner is inexpensive, available in most work offices, and easier to operate by non-skilled users. The appropriate instrumental conditions for measuring were selected using a genetic algorithm (GA) coupled with partial least square (PLS) regression. Using the conditions selected by GA, PLS and multiple linear regression (MLR) were compared, and similar results for the two methods were obtained. Under the selected conditions for each of the colorants, artificial neural network (ANN) including three layers of nodes and a Levenberg-Marquardt learning rule was employed, which improved the results. The concentration ranges for the three colorants in the multivariate calibration models were 0.00–5.31mmolL^–1 for amaranth, 0.00–1.85mmolL^–1 for brilliant blue, and 0.00–21.57mmolL^–1 for tartrazine. The minimum estimated relative standard error percentages (RSE%) for prediction of analytes in synthetic samples, using ANN with optimized parameters, were 16.8% for amaranth, 4.8% for brilliant blue, and 5.6% for tartrazine. A number of commercial food products were analyzed satisfactorily with the proposed method.     

Electric-dipole allowed （El） and forbidden （E2, M1 and M2） transition probabilities of 4f for N＋

By applying systematically enlarged multi-configuration Dirac-Fock wavefunction, the transitions for electricdipole allowed （E1） and forbidden （E2, M1 and M2） lines are studied among 4f pair coupling and low-lying configurations for singly ionized nitrogen. Most important effects of relativity, electron correlation, the rearrangement of electron density, Breit interaction, and quantum electrodynamic effects are included in the computation. Then, allowed （E1） and forbidden （E2, M1 and M2） transition probabilities of 4f for N＋ are obtained and compared with experimental results. Good agreement with available experimental results is found and most of the data of 4f are presented for the first time.     

Forbidden sets and stability in some rational difference equations

In this paper, we solve open problem (5) submitted by Sedaghat in his paper, On third order rational difference equations with quadratic terms, J. Differ. Equ. Appl., 14(8) (2008), pp. 889–897. We also confirm conjecture (6) in the mentioned paper.     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Characterization of microcrystals in some ancient glass beads from china by means of confocal Raman microspectroscopy

A total of ten ancient colored glass beads were analyzed by confocal Raman microspectroscopy for the non‐destructive identification of microcrystals within them. These beads were excavated from different regions of China, including Xinjiang, Henan, Hubei and Guangxi Provinces, and were dated mainly from the 10^th century BC to the 9^th century AD. For the first time, either tin or antimony‐based opacifiers/colorants including calcium antimonate (CaSb₂O₆, Ca₂Sb₂O₇), bindheimite (Pb₂Sb₂O₇), lead tin yellow type II (PbSn_1‐xSi_xO₃) and cassiterite (SnO₂) were identified in nine samples. In addition, other crystalline phases such as cuprite (Cu₂O), α‐wollastonite (CaSiO₃), diopside (CaMgSi₂O₆), feldspar (KAlSi₃O₈), calcite (CaCO₃) and quartz (SiO₂) were also detected. Another interesting phenomenon first observed in this study was the coexistence of Sn‐ and Sb‐based opacifiers/colorants in one mosaic bead from Guangxi. The possibility to use Sb‐ and Sn‐based opacifiers/colorants for dating and provenance study of ancient glass found in China is discussed briefly. Copyright © 2013 John Wiley & Sons, Ltd.