On laplace continued fraction for the normal integral

The Laplace continued fraction is derived through a power series. It provides both upper bounds and lower bounds of the normal tail probability % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqbfA6agzaaraaaaa!3DC0!\[\bar \Phi\](x), it is simple, it converges for x>0, and it is by far the best approximation for x3. The Laplace continued fraction is rederived as an extreme case of admissible bounds of the Mills' ratio, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqbfA6agzaaraaaaa!3DC0!\[\bar \Phi\](x)/(x), in the family of ratios of two polynomials subject to a monotone decreasing absolute error. However, it is not optimal at any finite x. Convergence at the origin and local optimality of a subclass of admissible bounds are investigated. A modified continued fraction is proposed. It is the sharpest tail bound of the Mills' ratio, it has a satisfactory convergence rate for x1 and it is recommended for the entire range of x if a maximum absolute error of 10^-4 is required.The efforts of the author were supported by the NSERC of Canada.     

Superstable theories with few countable models

Summary We prove here: Theorem. LetT be a countable complete superstable non -stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG ^eq orthogonal to the generics has Morley rank. Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.Supported by NSF grant DMS 90-06628     

Photopolymerized monolithic capillary columns for rapid micro high-performance liquid chromatographic separation of proteins

The preparation of monolithic poly(butyl methacrylate-co-ethylene dimethacrylate) capillary columns using photoinitiated in situ polymerization within 200 microm i.d. capillaries and their application for microHPLC separations of proteins have been studied. The low resistance to flow characteristic of monolithic columns, enabled the use of very high flow rates of up to 100 microL/min representing a flow velocity of 87 mm/s. Very good separations of a model protein mixture consisting of ribonuclease A, cytochrome c, myoglobin, and ovalbumin was achieved in less than 40 s using a very simple single step gradient of the mobile phase. Interestingly, no effect of the pore size on the separations of proteins was observed for these monolithic columns within the size range of 0.66-2.2 microm. The monolithic microHPLC columns are found very robust and no changes in the long term separation performance and back pressure were observed.