Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

High-resolution NMR of static samples by rotation of the magnetic field

Mechanical rotation of a sample at 54.7 degrees with respect to the static magnetic field, so-called magic-angle spinning (MAS), is currently a routine procedure in nuclear magnetic resonance (NMR). The technique enhances the spectral resolution by averaging away anisotropic spin interactions thereby producing isotropic-like spectra with resolved chemical shifts and scalar couplings. It should be possible to induce similar effects in a static sample if the direction of the magnetic field is varied, e.g., magic-angle rotation of the B0 field (B0-MAS). Here, this principle is experimentally demonstrated in a static sample of solid hyperpolarized xenon at approximately 3.4 mT. By extension to moderately high fields, it is possible to foresee interesting applications in situations where physical manipulation of the sample is inconvenient or impossible. Such situations are expected to arise in many cases from materials to biomedicine and are particularly relevant to the novel approach of ex situ NMR spectroscopy and imaging.     

NMR in rotating magnetic fields: magic-angle field spinning

Magic-angle sample spinning is one of the cornerstones in high-resolution NMR of solid and semisolid materials. The technique enhances spectral resolution by averaging away rank 2 anisotropic spin interactions, thereby producing isotropic-like spectra with resolved chemical shifts and scalar couplings. In principle, it should be possible to induce similar effects in a static sample if the direction of the magnetic field is varied (e.g., magic-angle rotation of the B0 field). Here we will review some recent experimental results that show progress toward this goal. Also, we will explore some alternative approaches that may enable the recovery of spectral resolution in cases where the field is rotating off the magic angle. Such a possibility could help mitigate the technical problems that render difficult the practical implementation of this method at moderately strong magnetic fields.     

A geodesic framework for analyzing molecular similarities

A fast self-organizing algorithm for extracting the minimum number of independent variables that can fully describe a set of observations was recently described (Agrafiotis, D. K.; Xu, H. Proc. Natl. Acad. Sci.U.S.A. 2002, 99, 15869-15872). The method, called stochastic proximity embedding (SPE), attempts to generate low-dimensional Euclidean maps that best preserve the similarities between a set of related objects. Unlike conventional multidimensional scaling (MDS) and nonlinear mapping (NLM), SPE preserves only local relationships and, by doing so, reveals the intrinsic dimensionality and metric structure of the data. Its success depends critically on the choice of the neighborhood radius, which should be consistent with the local curvature of the underlying manifold. Here, we describe a procedure for determining that radius by examining the tradeoff between the stress function and the number of connected components in the neighborhood graph and show that it can be used to produce meaningful maps in any embedding dimension. The power of the algorithm is illustrated in two major areas of computational drug design: conformational analysis and diversity profiling of large chemical libraries.     

Robust discrete optimization and network flows

We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation. When only the cost coefficients are subject to uncertainty and the problem is a 0–1 discrete optimization problem on n variables, then we solve the robust counterpart by solving at most n+1 instances of the original problem. Thus, the robust counterpart of a polynomially solvable 0–1 discrete optimization problem remains polynomially solvable. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. are polynomially solvable. We also show that the robust counterpart of an NP-hard -approximable 0–1 discrete optimization problem, remains -approximable. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network. The research of the author was partially supported by the Singapore-MIT alliance.The research of the author is supported by a graduate scholarship from the National University of Singapore.Mathematics Subject Classification (2000): 90C10, 90C15     

Polymerization beyond the gel point. I. The molecular weight of sol as a function of the extent of reaction

Theoretical expressions for describing the weight-average molecular weights of the soluble fractions from polymerizations obtained beyond the gel point were tested experimentally. The theory of branching processes and a recursive approach essentially based on the method of Macosko and Miller were found to be virtually equivalent. The soluble fractions produced from the stoichiometric polymerization of 1,3,5-benzenetriacetic acid (BTA) with decamethylene glycol (DMG) gave molecular weights and distributions in excellent qualitative agreement with the theory. The results are interpreted in terms of intramolecular cyclization, diffusion, and the presence of microgel particles.     

Isomorphism within the hexagonal columnar mesophase of molecular and macromolecular self- and co-assembled columns containing tapered groups

The phase behaviour of binary mixtures of self-assembled tapering molecules based on monoesters of oligooxyethylene glycol and 3,4,5-tris[4-(n-dodecan-1-yloxy)benzyloxy]benzoic acid, their corresponding polymethacrylates, and of 4'-methyl (benzo-15-crown-5)-3,4,5-tris[4-(n-dodecan-1-yloxy)benzyloxy]benzoate within their hexagonal columnar mesophase (Φ_h) is described. The binary blends between molecular tapers co-assemble into a single supramolecular column resulting in isomorphism within their Φ_h mesophase over the entire range of composition. The binary blends between polymethacrylates containing tapered side groups co-assemble into a single Φ_h phase only when the columns of the parent polymers are of similar diameters. This results in binary mixtures which are isomorphic within the Φ_h mesophase over the entire composition range. When the diameters of the columns formed by the parent polymers are dissimilar, isomorphic mixtures are obtained only over a narrow range of composition. Binary mixtures between molecular tapers and macromolecular systems containing tapered side groups co-assemble into a single column to the extent that intercalation of the molecular taper, within the column formed by the macromolecular system containing tapered side groups, is permissible. In all systems increased intracolumnar interactions can be induced by complexation of CF₃SO₃Li by the oligooxyethylenic receptors leading to isomorphism in otherwise non-isomorphic mixtures. Ternary mixtures between molecular tapers with non-specific oligooxyethylenic receptors and specific crown ether receptors and CF₃SO₃ Na as the third component are non-isomorphic within their Φ_h phase due to preferential complexation of the alkali metal cation by the column of the crown ether containing the molecular taper. This results in two columns of dissimilar diameters, which are isomorphic in the Φ_h phase only within a limited range of composition.     

Regularity properties of distributions through sequences of functions

We give necessary and sufficient criteria for a distribution to be smooth or uniformly Hölder continuous in terms of approximation sequences by smooth functions; in particular, in terms of those arising as regularizations ${(T\ast\phi_{n})}$ .     

On the approximability of adjustable robust convex optimization under uncertainty

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.