On quadratic fields with large 3-rank

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the general cubic discriminants (real or imaginary) up to in time and space , or more generally in time and space for a freely chosen positive . A variant computes the -ranks of all quadratic fields of discriminant up to with the same time complexity, but using only units of storage. As an application we obtain the first real quadratic fields with , and prove that is the smallest imaginary quadratic field with -rank equal to .

     

High‐throughput characterization of pattern formation in symmetric diblock copolymer films

Surface‐pattern formation in thin block copolymer films was investigated by utilizing a high‐throughput methodology to validate the combinatorial measurement approach and to demonstrate the value of the combinatorial method for scientific investigation. We constructed measurement libraries from images of subregions of block copolymer films having gradients in film thickness and a range of molecular mass, M. A single gradient film covers a wide range of film morphologies and contains information equivalent to a large number of measurements of films having a fixed thickness, h. Notably, the scale of the surface patterns is generally much larger than the molecular dimensions so that the interpretation of the patterns is more subtle than ordering in bulk block copolymer materials, and there is no predictive theory of this type of surface‐pattern formation. We observed a succession of surface patterns that repeat across the film with increasing h [extended smooth regions, regions containing circular islands, labyrinthine (“spinodal”) patterns, holes, and smooth regions again]. The extended smooth regions and the labyrinthine patterns appear to be novel features revealed by our combinatorial study, and these patterns occurred as bands of h that were quantized by integral multiples of the bulk lamellar period, L_o. The magnitude of the height gradient influenced the width of the bands, and the smooth regions occupied an increasing fraction of the film‐surface area with an increasing film gradient. The average size of the spinodal patterns, λ, was found to scale as λ ～ L or λ ～ M^?1.65 and reached a limiting size at long annealing times. The hole and island features had a size comparable to λ, and their size likewise decreased with increasing M. The smooth regions were attributed to an increase in the surface‐chain density in the outer brushlike block copolymer layer with increasing h, and the scaling of λ with M was interpreted in terms of the increasing surface elasticity with M. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2141–2158, 2001     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

Role of interfacial chemistry on the rheology and thermo-mechanical properties of clay-polymer nanocomposites for building applications

This study is directed towards investigating the role of the surface treatment of clay particles on the rheological and thermomechanical behaviour of clay-epoxy blends. Nanocomposites were prepared by mixing small amounts (5–10 mass %) of commercial organoclays or raw clays with an epoxy system commonly used in civil engineering. Rheological characterisations in the liquid state revealed a pronounced thixotropic character of the organoclay-based systems, which all exhibited a shear-thinning behaviour above a critical stress threshold (yield stress), depending on both the intensity of interfacial interactions and the degree of filler dispersion. On the other hand, systems based on raw clay particles behaved like Newtonian fluids, in the same way as the unreinforced polymer matrix. Complementary dynamic mechanical analyses (DMA) performed on the cured cross-linked nanocomposites also showed significant changes in the viscoelastic behaviour of the epoxy matrix due to the introduction of organoclays, whereas only minor variations were observed following the introduction of raw fillers. These results were consistent with nanoscale morphological characterisations performed by conventional X-ray diffraction (XRD) on the various hybrid systems. In this context, rheology and DMA appear as attractive alternative methods for assessing the filler dispersion at a macroscopic (and possibly more relevant) scale. This research is of practical interest for civil engineers, since clay reinforced-epoxies could in the future be used as coating materials with enhanced barrier performances, in order to protect infrastructures against environmental ageing or corrosion.     

Varying Track Etch Rates along the Fission Fragments＇ Trajectories in CR-39 Detectors

We determine the behavior of track etch rate VT along the tracks in CR-39 detectors exposed to fission fragments from ^252Cf source. CR-39 detectors are etched at 70℃ in different concentrations of Na2COa-mixed 6M NaOH solutions for different etching time intervals of 5-10min starting from 15 min up to 210min. Two values of track etch rate are determined along the fission fragment trajectories. The variation in VT is correlated with the energy loss rate of the fission fragment in the detector material.     

Two local search approaches for solving real-life car sequencing problems

The NP-hard problem of car sequencing appears as the heart of the logistic process of many car manufacturers. The subject of the ROADEF’2005 challenge addressed a car sequencing problem proposed by the car manufacturer RENAULT, more complex than the academic problem generally addressed in the literature. This paper describes two local search approaches for this problem. In the first part, a new approach by very large-scale neighborhood search is presented. This approach, designed during the qualification stage preceding the final, is based on an original integer linear programming formulation. The second part is dedicated to the approach which enabled us to win the ROADEF’2005 challenge. Inspired by the latest works on the subject, this one is based on very fast explorations of small neighborhoods. Our contribution here is mainly algorithmic, in particular by showing how much exploiting invariants speeds up the neighborhood evaluation and contributes to the diversification of the search. Finally, the two approaches are compared and discussed through an extensive computational study on RENAULT’s benchmarks. The main conclusion drawn at this point is that sophisticated metaheuristics are useless to solve car sequencing problems. More generally, our victory on ROADEF’2005 challenge demonstrates that algorithmic aspects, sometimes neglected, remain the key ingredients for designing and engineering high-performance local search heuristics.     

CONTINUUM TOPOLOGY OPTIMIZATION FOR MONOLITHIC COMPLIANT MECHANISMS OF MICRO-ACTUATORS

A multi-objective scheme for structural topology optimization of distributed compliant mechanisms of micro-actuators in MEMS condition is presented in this work, in which mechanical flexibility and structural stiffness are both considered as objective functions. The compliant micro-mechanism developed in this way can not only provide sufficient output work but also have sufficient rigidity to resist reaction forces and maintain its shape when holding the work-piece. A density filtering approach is also proposed to eliminate numerical instabilities such as checkerboards, mesh-dependency and one-node connected hinges occurring in resulting mechanisms. SIMP is used as the interpolation scheme to indicate the dependence of material modulus on element-regularized densities. The sequential convex programming method, such as the method of moving asymptotes (MMA), is used to solve the optimization problem. The validation of the presented methodologies is demonstrated by a typical numerical example.