Perspectives on Advanced Mathematical Thinking

This article sets the stage for the following 3 articles. It opens with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of the International Group for the Psychology of Mathematics Education. It then locates the articles within 4 recurring themes: (a) the distinction between identifying kinds of thinking that might be regarded as advanced at any grade level, and taking as advanced any thinking about mathematical topics considered advanced; (b) the utility of characterizing such thinking for integrating the entire curriculum; (c) general tests, or criteria, for identifying advanced mathematical thinking; and (d) an emphasis on advancing mathematical practices. Finally, it points out some commonalities and differences among the 3 following articles.     

Large Scale Linear Programs and Heuristics for the Design of Survivable Telecommunication Networks

We consider the problem of providing a minimum cost multi-service network subject to one link failure scenarios. We assume our network is fully meshed and demand is satisfied by using direct or two hop-paths. We provide a large scale linear programming formulation and propose and test two effective heuristics.     

GC-MS and GC-MS-MS analysis of a complex essential oil

Summary The alcoholic portion of avetiver oil (Bourbon), the gas chromatogram of which shows more than 100 components, was analyzed by combined GC-MS with different ionization methods. This enabled us to identify major components. The specificity of the method was improved by using combined gas chromatography-tandem mass spectrometry in the collision-activated-dissociation mode.     

Direct determination of the enantiomeric purity of oxyphenonium using chiral HPLC with post-column extraction detection

Summary A method is described for the determination of the enantiomeric purity (enantiomeric excess) of the anticholinergic drug oxyphenonium. The method for this quaternary ammonium compound is based on the direct HPLC analysis with a chiral stationary phase. Two kinds of 1-acid glycoprotein-bonded phases were used.For the detection a post-column extraction with fluorescence detection of the ion-pair counter ion dimethoxyantracene sulphonate was used.     

Etude des propriétés cristallographiques,diélectriques et d'optique non linéaire de quelques nouvelles phases de composition BaxLi5−2xNb5(1−y)Ta5yO15 et de structure bronzes oxygénés de tungstène quadratiques

Two series of phases with tetragonal bronze-like structure and composition Ba_xLi_5?2xT₅O₁₅ (T = Nb, Ta) have been isolated in the systems BaNb₂O₆LiNbO₃ and BaTa₂O₆LiTaO₃. All these phases show ferroelectric-paraelectric transitions. The Curie temperature increases with the lithium content. The value of T_C for Ba_2.03Li_0.94Nb₅O₁₅ is the highest ever observed for this type of structure: the obtained phases are potentially good materials for the harmonic generation of the 0.53-μm radiation. The optical yield of the niobate Ba_2.14Li_0.71Nb₅O₁₅ is about 2.5 times that of Ba₂NaNb₅O₁₅ and 250 times that of the K.D.P. The crystallographic and dielectric data of the system Ba_2.14Li_0.71Nb₅O₁₅Ba_2.14Li_0.71Ta₅O₁₅ characterize three domains, which are respectively antiferroelectric, ferroelectric, and paraelectric. The Curie temperature and the optical yield decrease with increasing tantalum content.     

Supernovae and the Arrow of Time

Supernovae are explosions of stars and are a central problem in astrophysics. Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities develop during the star’s explosion and lead to intense interfacial RT/RM mixing of the star materials. We handle the mathematical challenges of the RT/RM problem based on the group theory approach. We directly link the conservation laws governing RT/RM dynamics to the symmetry-based momentum model, derive the model parameters, and find the analytical solutions and characteristics of RT/RM dynamics with variable accelerations in the linear, nonlinear and mixing regimes. The theory outcomes explain the astrophysical observations and yield the design of laboratory experiments. They suggest that supernova evolution is a non-equilibrium process directed by the arrow of time.     

Time reversal for infinite-dimensional diffusions

Summary In this paper we analize the reversibility of the diffusion property for the solution of certain infinite-dimensional systems of stochastic differential equations. Necessary and sufficient conditions ensuring this reversibility are given. The proofs use the techniques of the stochastic calculus of variations.This work was partly done when the first author was visiting the Centre de Recerca Matemàtica at Barcelona     

Rational approximation of vertical segments

In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.     

Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

The purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.