A Two-Parameter Family of Orthogonal Polynomials with Respect to a Jacobi-Type Weight on the Unit Circle

A two-parameter family of polynomials is introduced by a recursion formula. The polynomials are orthogonal on the unit circle with respect to the weight ω_{α, β}(θ) = |(1 − z)^α(1 + z)^β|², α, β > − , z = e^iθ. Explicit representation, norm estimates, shift identities, and explicit connection to Jacobi polynomials on the real interval [−1, 1] is presented.     

Incorporating the potential for human error in maintenance models

The mathematics of delay-time modelling of inspection maintenance is extended to incorporate the existence of human error in the form of fault injection during the inspection process. After briefly discussing the basic delay-time model, modifications are introduced to model maintenance scenarios incorporating human error injected defects within the inspection maintenance process. The effects of human error are investigated with the emphasis on its representation and on the assessment of consequences, the objective being to provide a means of determining the cost of human error and to thereby aid corrective decision-making.     

A periodic testing model for a preparedness system with a defective state

This paper considers the periodic testing of a preparednesssystem where in addition to working and failed state recognition,a working but defective state also exists. Based upon the delaytime model, an expected availability model is derived and evaluatedas a function of the constant inspection period. The model enablesthe range of inspection periods which satisfy a pre-set availabilitycriterion to be established, and the optimal availability inspectionperiod to be identified. Variants of the basic model are considered including: wherea delay time period exists, but the technology to detect a defectis not available; where the delay time is zero, so that onlyfailures are detected; and where the system is replaced on aregular basis without any state testing. These variants enablethe value and effectiveness of the ability to detect defectsand to detect failures to be identified and quantified. The models are demonstrated in the context of a missile buffersystem, where the numerical example clarifies the value of modellingand the insight into the potential effectiveness of defect andfailure detection.     

A semigroup of operators in convexity theory

We consider three operators which appear naturally in convexity theory and determine completely the structure of the semigroup generated by them.

RESUMO. Duongrupo de operatoroj en la teorio pri konvekseco. Ni konsideras tri operatorojn kiuj aperas nature en la teorio pri konvekseco kaj plene determinas la strukturon de la duongrupo generita de ili.

     

Delay-Time Model of Reliability of Equipment Subject to Inspection Monitoring

A model is developed for the reliability of a single component, subject to one type of inspectable defect, which will subsequently lead to a failure. Inspections are assumed perfect. The model utilizes the notion of delay time to establish the reliability consequences of inspecting on different inspection periods. The argument is readily extended to the case of a component with multiple and independent defects. Numerical examples are provided.     

Monitoring of Saccharomyces cerevisiae cell proliferation on thiol-modified planar gold microelectrodes using impedance spectroscopy

An impedance spectroscopic study of the interaction between thiol-modified Au electrodes and Saccharomyces cerevisiae of strain EBY44 revealed that the cells formed an integral part of the interface, modulating the capacitive properties until a complete monolayer was obtained, whereas the charge transfer resistance ( R ct) to the redox process of [Fe(CN)6] 3-/4- showed a linear relationship to the number of cells even beyond the monolayer coverage. R ct showed strong pH dependence upon increasing the pH of the utilized buffer to 7.2. Upon addition of S. cerevisiae cells at pH 7.2, the obtained value of R ct showed over 560% increase with respect to the value obtained on the same thiol-modified electrode without cells. It was demonstrated that real-time monitoring of S. cerevisiae proliferation, with frequency-normalized imaginary admittance (real capacitance) as the indicator, was possible using a miniaturized culture system, ECIS Cultureware, with integrated planar cysteamine-modified Au microelectrodes. A monolayer coverage was reached after 20-28 h of cultivation, observed as an approximately 15% decrease in the real capacitance of the system.     

Inequalities of the Brunn–Minkowski type for Gaussian measures

Let be an integer, let γ be the standard Gaussian measure on , and let . Given this paper gives a necessary and sufficient condition such that the inequality is true for all Borel sets A ₁,...,A _m in of strictly positive γ-measure or all convex Borel sets A ₁,...,A _m in of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn–Minkowski type for γ which are true for all convex sets but not for all measurable sets.      

Bridging-cluster model for hydrophobic attraction

A new model is proposed to account for the long-range hydrophobic attraction repeatedly observed for thin water films between two stable (solid) hydrophobic surfaces. The model is based on the notion of structurally organized, elongated water clusters that span the gap between the hydrophobic surfaces. Two features are noted: (i) Mixing entropy due to the mixing of the clusters and the remainder of the water in the thin film is explicitly taken into account. (ii) A term is invoked that depends inversely on the film thickness, which accounts for the free-energy change associated with reorganizing the film as the film thickness varies. Fitting to experimental surface force data resulted in parameter values of reasonable magnitudes. The model developed covers film thicknesses from about 2 nm and above. On this basis, the amazingly long range of the hydrophobic attraction can be attributed to the formation of bridging, quasi-cylindrical clusters having a radius on the order of 1 to 2 nm.     

The Impact of Halogen Substituents on the Synthesis and Structure of Co-Crystals of Pyridine Amides

Strategies for co-crystal synthesis tend to employ either hydrogen- or halogen-bonds between different molecules. However, when both interactions are present, the structural influence that they may exert on the resulting assembly is difficult to predict a priori. To shed some light on this supramolecular challenge, we attempted to co-crystallize ten aliphatic dicarboxylic acids (co-formers) with three groups of target molecules; N-(pyridin-2-yl)picolinamides (2Pyr-X), N-(pyridin-2-yl)nicotinamides (3Pyr-X), N-(pyridin-2-yl)isonicotinamides (4Pyr-X); X=Cl/ Br/ I. The structural outcomes were compared with co-crystals prepared from the non-halogenated targets. As expected, none of the reactions with 2Pyr-X produced co-crystals due to the presence of a very stable intramolecular N-H···N hydrogen bond. In the 3Pyr series, all six structures obtained showed the same synthons, –COOH···N(py) and –COOH···N(py)-NH, that were found in the non-halogenated parent 3Pyr and were additionally accompanied by structure directing X···O(OH) interactions (X=Br/I). The co-crystals of the unhalogenated parent 4Pyr co-crystals assembled via intermolecular –COOH···N(py) and –COOH···N(py)-NH synthons. Three of the analogues 4Pyr-X co-crystals displayed only COOH···N(py) and –COOH···N(py)-NH interactions. The three co-crystals of 4Pyr-X with fumaric acid (for which no analogues structures with 4Pyr are known) formed –COOH···N(py)-NH and –NH···O=C hydrogen bonds and showed no structure-directing halogen bonds. In three co-crystals of 4Pyr-I in which –COOH···N(py)-NH hydrogen bond was present, a halogen-bond based –I···N(py) synthon replaced the –COOH···N(py) motif observed in the parent structures. The structural influence of the halogen atoms increased in the order of Cl < Br < I, as the size of σ-holes increased. Finally, it is noteworthy that isostructurality among structures of the homomeric targets was not translated to structural similarities between their respective co-crystals.