Natural frequencies of two bubbles in a compliant tube: analytical, simulation, and experimental results

Motivated by various clinical applications of ultrasound contrast agents within blood vessels, the natural frequencies of two bubbles in a compliant tube are studied analytically, numerically, and experimentally. A lumped parameter model for a five degree of freedom system was developed, accounting for the compliance of the tube and coupled response of the two bubbles. The results were compared to those produced by two different simulation methods: (1) an axisymmetric coupled boundary element and finite element code previously used to investigate the response of a single bubble in a compliant tube and (2) finite element models developed in comsol Multiphysics. For the simplified case of two bubbles in a rigid tube, the lumped parameter model predicts two frequencies for in- and out-of-phase oscillations, in good agreement with both numerical simulation and experimental results. For two bubbles in a compliant tube, the lumped parameter model predicts four nonzero frequencies, each asymptotically converging to expected values in the rigid and compliant limits of the tube material.     

Controlled synthesis and functionalization of PEGylated methacrylates bearing cyclic carbonate pendant groups

Homopolymer bearing cyclic carbonate (CC) group, ABA type triblock copolymers, and (AC)B(AC) type terpolymers with statistical arrangement of A and C monomers bearing side chain CC groups are reported here. Difunctional poly(ethylene glycol) macroinitiators (PEGMIs) were prepared from PEG of three different molecular weights. PEGMIs were subsequently used for the preparation of polymers bearing CC pendant groups from cyclic carbonate methacrylate (CCMA) under atom transfer radical polymerization to yield polymers with low polydispersity index. Homopolymer and ABA type triblock copolymers were obtained by polymerizing CCMA monomer and (AC)B(AC) type statistical terpolymers were obtained when methyl methacrylate was included as a comonomer. No polymer was obtained when styrene was used as comonomer. The cyclic carbonate groups were subjected to ring‐opening reaction with monoamine to yield side chain hydroxyurethane polymers with increased solubility and diamines to yield crosslinked insoluble materials. Changes in wettability characteristics were studied by following the water contact angle of the polymers before and after ring‐opening reaction involving the cyclic carbonate pendant group. The polymers which composed of electrolyte in the form of PEG and coordinating species in the form of pendant cyclic carbonate groups showed conductivity in the range of 2–5 × 10^?6 Scm^?1 at 23 °C after doping with lithium bis(trifluoromethane)sulfonimide as characterized by impedance spectroscopy. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 1622–1632, 2010     

Supersymmetric Higgs boson decays in the MSSM with explicit <Emphasis Type="Italic">CP</Emphasis> violation

Decays into neutralinos and charginos are among the most accessible supersymmetric decay modes of Higgs particles in most supersymmetric extensions of the standard model. In the presence of explicitly CP violating phases in the soft breaking sector of the theory, the couplings of Higgs bosons to charginos and neutralinos are in general complex. Based on a specific benchmark scenario of CP violation, we analyze the phenomenological impact of explicit CP violation in the minimal supersymmetric standard model on these Higgs boson decays. The presence of CP violating phases could be confirmed either directly through the measurement of a CP odd polarization asymmetry of the produced charginos and neutralinos, or through the dependence of CP even quantities (branching ratios and masses) on these phases. Received: 18 April 2002 / Published online: 26 July 2002     

Letter to the editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor- in- chief, Chandler Davis.     

Particle physics explanations for ultra-high energy cosmic ray events

The origin of cosmic ray events withE ≳ 10¹¹ GeV remains mysterious. In this talk I briefly summarize several proposed particle physics explanations: a breakdown of Lorentz invariance, the ‘Z-burst’ scenario, new hadrons with masses of several GeV as primaries, and magnetic monopoles with mass below 10¹⁰ GeV as primaries. I then describe in a little more detail the idea that these events are due to the decays of very massive, long-lived exotic particles.     

Synthesis of 2,3-diaminodihydropyrroles via thioimidate cyclopropane rearrangement

The synthesis of novel 2,3-diaminodihydropyrroles is reported. The key step is a thiomethylimidate cyclopropane rearrangement to afford a pyrrolothiomethylimidate intermediate. Reaction of this intermediate with amines produced the title compounds in good yields. The scope and mechanism of the reaction is discussed.     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Transformations of 2,2-dimethyl-2,4-dihydro-3H-pyrrol-3-on-1-oxide derivatives in the Vilsmeier-Haack reaction conditions

Enhydroxylaminones—2,2-dimethyl-2,4-dihydro-3H-pyrrol-3-on-1-oxides were shown to give various chlorinated products in the Vilsmeier-Haack reaction. The general sequence of the reaction steps is determined and the extent of the reaction was shown to be strongly dependent on the substrate structure.