The enumerative geometry of surfaces and modular forms

Let be a surface, and let be a holomorphic curve in representing a primitive homology class. We count the number of curves of geometric genus with nodes passing through generic points in in the linear system for any and satisfying .

When , this coincides with the enumerative problem studied by Yau and Zaslow who obtained a conjectural generating function for the numbers. Recently, Göttsche has generalized their conjecture to arbitrary in terms of quasi-modular forms. We prove these formulas using Gromov-Witten invariants for families, a degeneration argument, and an obstruction bundle computation. Our methods also apply to blown up at 9 points where we show that the ordinary Gromov-Witten invariants of genus constrained to points are also given in terms of quasi-modular forms.     

Experimental study of optimum spacing problem in the cooling of simulated electronic package

  An experimental investigation has been performed to determine the effects of different arrangements of obstacles on the cooling of simulated electronic package. The considered simulated electronic package consisted of a channel formed by two parallel plates. The bottom plate is attached with five identical electrically heated square obstacles, which are perpendicular to the mean airflow and arranged with different side-to-side distances. The experimental results show that the conventional equi-spaced arrangement might not be the optimum option and should be avoided. A better thermal performance could be obtained when the side-to-side distances between the obstacles followed a geometric series. For example, at Re=800, the highest temperature of the optimum arrangement could be reduced by 12% compare to the equi-spaced arrangement and the maximum temperature difference among the five obstacles is lower than that of equi-spaced arrangement by 32.1%. Received on 17 December 1999     

Metal-catalyzed cyclopropanation on the 8-oxabicyclo[3.2.1]oct-6-ene template

Cyclopropanations of an 8-oxabicyclo[3.2.1]octene substrate using diazocarbonyl compounds provided exo, exo-cyclopropanated products as the sole or major diastereomeric oxatricyclic products. Reductive cleavage of a meso-oxatricyclic ketone by samarium iodide resulted in desymmetrization without concomitant oxygen bridge cleavage.     

Characterization of Kovar-to-Kovar laser welded joints and its mechanical strength

For the packaging of a pump laser in butterfly package, the most crucial assembly step is the fiber-to-laser diode coupling and attachment. The use of laser welding as the joining method offers several advantages if compared with the adhesive joints: strong joining strength, short process time and less contamination. This paper reports on laser welding process characteristics; weld strength and its fracture mode. The penetration depth and melt area of laser spot welds were found to be complicated functions of laser pulse energy, intensity, and beam diameter. Effects of pulse width, input power and size of the focal spot on the rate of energy input to the workpieces and consequently, the weld strength were reported. The weld strength was found to be dependent on the overlapping area between the two joining materials. Surface roughness, R_a, has influence on the fraction of energy absorbed, A, and therefore, affecting the penetration depth. Thermal analysis was carried out on the laser-welded joints and its heat-affected zone (HAZ) induced by various power densities was examined. These data are important in order to optimize and utilize the laser welding process as an effective manufacturing tool for fabrication of reliable pump laser.     

Future teachers’ professional knowledge on argumentation and proof: a case study from universities in three countries

In this paper, qualitative results of a case study about the professional knowledge in the area of argumentation and proof of future teachers from universities in three countries are described. Based on results of open questionnaires, data about the competencies these future teachers have in the areas of mathematical knowledge and knowledge of mathematics pedagogy are presented. The study shows that the majority of the future teachers at the participating universities situated in Germany, Hong Kong and Australia, were not able to execute formal proofs, requiring only lower secondary mathematical content, in an adequate and mathematically correct way. In contrast, in all samples there was evidence of at least average competencies of pedagogical content reflection about formal and pre-formal proving in mathematics teaching. However, it appears that possessing a mathematical background as mandated for teaching and having a high affinity with proving in mathematics teaching at the lower secondary level are not a sufficient preparation for teaching proof.     

Determination of vitamin U and its degradation products by high- performance liquid chromatography with fluorescence detection

A high-performance liquid chromatographic method has been developed for the determination of vitamin U in tablets and capsules. Threonine was employed as the internal standard through the assay. The o-phthalaldehyde derivatives were prepared and then chromatographed isocratically on a reversed-phase C18 column. The optimum reaction time for both vitamin U and threonine at pH 10.5 is 5 min. Vitamin U and its major degradation product in the dosage forms, viz., methionine sulphoxide, were separated and quantified with a relative standard deviation of about 1%, using a fluorescence detector with excitation and emission wavelengths at 340 and 450 nm respectively. A linear relationship has been established between the peak area ratio of vitamin U/threonine and the concentration of vitamin U over the range of 2.5-50 micrograms/ml.     

Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p Close to 0

Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for p > 1, we demonstrate here by a careful random selection argument that also at p = 0, and consequently for sufficiently small p, there exist counterexamples. An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≲ 0.11. We conjecture however that violations of additivity exist for all p < 1.     

Scheduling orders on either dedicated or flexible machines in parallel to minimize total weighted completion time

We are interested in the problem of scheduling orders for different product types in a facility with a number of machines in parallel. Each order asks for certain amounts of various different product types which can be produced concurrently. Each product type can be produced on a subset of the machines. Two extreme cases of machine environments are of interest. In the first case, each product type can be produced on one and only one machine which is dedicated to that product type. In the second case, all machines are identical and flexible; each product type can be produced by any one of the machines. Moreover, when a machine in this case switches over from one product type to another, no setup is required. Each order has a release date and a weight. Preemptions are not allowed. The objective is minimizing the total weighted completion time of the orders. Even when all orders are available at time 0, both types of machine environments have been shown to be NP-hard for any fixed number (≥2) of machines. This paper focuses on the design and analysis of approximation algorithms for these two machine environments. We also present empirical comparisons of the various algorithms. The conclusions from the empirical analyses provide insights into the trade-offs with regard to solution quality, speed, and memory space. Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users. This research is supported by the National Science Foundation through grants DMI-0300156 and DMI-0245603.     

Synthesis of N-benzyl-des-D-ring lamellarin K via an acyl-Claisen/Paal-Knorr approach

Lamellarin K is a complex pyrrole natural product and member of the lamellarin family – a group of natural products known for their potent biological activities, such as, antiproliferative activity and inhibition of P-gp mediated drug efflux pumps. We herein describe the synthesis of the N-benzyl-des-D ring analogue of lamellarin K using a route that centres on an acyl-Claisen reaction to eventually prepare a highly-functionalised 1-aryl-4-methyl-1,4-diketone. Paal-Knorr pyrrole formation using this diketone undergoes auto-oxidation to give a fully-substituted 5-formyl pyrrole which was converted into the natural lactone B ring. Antiproliferative testing of the N-benzyl-des-D ring analogue gave an IC₅₀ of 2.63 μM against the MDA-MB-231 breast cancer cell line.