On problematic aspects in learning trigonometry

In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed.

The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle.     

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

The problem of computing oscillatory integrals with general oscillators is considered. We employ a Filon-type method, where the interpolation basis functions are chosen in such a way that the moments are in terms of elementary functions and the oscillator only. This allows us to evaluate the moments rapidly and easily without needing to engage hypergeometric functions. The proposed basis functions form a Chebyshev set for any oscillator function even if it has some stationary points in the integration interval. This property enables us to employ the Filon-type method without needing any information about the stationary points if any. Interpolation by the proposed basis functions at the Fekete points (which are known as nearly optimal interpolation points), when combined with the idea of splines, leads to a reliable convergent method for computing the oscillatory integrals. Our numerical experiments show that the proposed method is more efficient than the earlier ones with the same advantages.     

Design,Synthesis, and Anticancer Activity of New Oxadiazolyl‐Linked and Thiazolyl‐Linked Benzimidazole Arylidines,Thioglycoside, and Acyclic Analogs

2‐[(4‐Thiazolylmethyl)thio]‐1H‐benzimidazole 3 was prepared and was allowed to react with ethyl chloroactate then with hydrazine hydrate to afford the hydrazide derivative 5 , which was then reacted with aromatic aldehydes to afford the corresponding arylidine derivatives 6 – 9 . Heterocyclization of the latter hydrazones with acetic anhydride afforded the substituted 1,3,4‐oxadiazoline derivatives 10 – 13 . In addition, new ((thiazolyl)imidazolyl) oxadiazole thioglycoside and acyclic‐C nucleoside analog were prepared via heterocylization of the hydrazide 5 then glycosylation with α‐acetobromoglucose or condensation with D‐xylose, respectively. All the new compounds were structurally characterized. The anticancer activity of some of the newly synthesized compounds was studied against human breast cancer cells (MCF‐7). The results of the anticancer activity showed that compounds 8 , 11 , 12 , 17 , and 18 revealed high activities superior to Doxorubicin; however, the other derivatives showed moderate to low inhibition activities against human breast cancer cells. Docking studies into CDK2 enzyme were investigated, which supported the anticancer activity results.     

On the Sensitiveness of Absorption Coefficient to Changes in Material Parameters

An essential property of porous absorbers is the resistance to the incident sound wave, resulting in a damping effect. Damping results from the friction between air molecules and absorber pore-walls. In practice, the known dependencies of the sound absorption coefficient are used in the design of absorbers. But not every material can be characterized by a one-parameter model. The reliability of prediction can be improved by using detailed material models. Hence, additional material parameters, especially the pore geometry are needed. Since in practice the effort for determining material parameters must be minimized, this paper discusses the sensitivity of prediction of absorbers' performance depending on the influence of material parameters that are input for the prediction model chosen. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Kepler problem spinor regularization and pre-quantization of the negative-energy manifold

Summary We extend the results of our two foregoing papers to the whole phase space of the Kepler problem, by: i) developing a natural spinor extension of the KS-regulari-zation of the Kepler problem, due to Kustaanheimo and Stiefel, and ii) applying the results to the pre-quantization (in the sense of Souriau) of the negative-energy manifold, the energy levels satisfying a well-known condition. We show that the quantized manifold is diffeomorphic to the symplectic productS ₃×S ₃ of two 3-spheres of the spinor phase space, factored through a suitable equívalence relation.

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Sommario Estendendo i risultati ottenuti in due prece denti lavori all'intero spazio delle fasi del problema di Keplero, si perviene nella presente nota ad una naturale estensione spinorialedella trasformazione KS (introdotta da Kustaanheimo e Stiefel per regolarizzare il problema di Keplero) la quale consente di prequantizzare, nel senso di Souriau, la varietà delle orbite kepleriane isoenergetiche, con energia negativa spddisfacente ad una ben nota condizione. Si mostra che la varietà quantica associata è diffeomorfa al prodotto simplettico S ₃×S ₃,opportunamente quozientato, di due 3-sfere dello spazio spinoriale delle fasi.

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Work performed under the auspices of G.N.F.M. of the Italian Research Council (C.N.R.).     

Synthesis of novel 5,6,7,8,9,10-hexahydropyrimido[4,5-b]quinoline derivatives for antimicrobial and anti-oxidant evaluation

Because of the well-known chemotherapeutic activity of pyrimidoquinolines, a new series of the title compounds were synthesized and biologically screened for their antimicrobial and anti-oxidant activities. The intermediate compound, 2-amino-1-cyclohexyl-4-(3,4-dimethoxyphenyl)-1,4,5,6,7,8-hexahydroquinoline-3-carbonitrile (2), was prepared and allowed to react with oxalyl chloride to give the corresponding strategic starting material, 10-cyclohexyl-5-(3,4-dimethoxyphenyl)-4-oxo-3,4,5,6,7,8,9,10-octahydropyrimido[4,5-b]quinoline-2-carbonyl chloride 3. Compound 3 underwent diverse reactions with amines to give the corresponding carboxylic amides 4, with alcohols to give the carboxylic esters 5 and with hydrazine to give the carbohydrazide 8. Compound 8 reacted with aldoses to give the corresponding polyhydroxy(-)alkyl Schiff bases 9, which upon reaction with thioglycolic acid afforded the thiazolidinone-C-acyclic nucleosides 10. Several other pyrimidoquinolines incorporated to oxadiazole, iminothiazolidinones and thiadiazoles were synthesized starting from the 2-carbohydrazide derivative 8. Most of the newly prepared derivatives showed considerable antimicrobial activity against Gram +ve and Gram ?ve bacteria and anti-oxidant activity.     

New weighing matrices of weight 25

The existence of a weighing matrix of order 33 and weight 25 has been open so far. We actually construct such a circulant matrix, thereby obtaining circulant matrices of order 33t with weight 25, for each positive integer t. Consequently a missing entry in Craigen's table of weighing matrices can now be filled with a positive response. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 11–15, 1999