Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.Comment: 11 pages, 4 figure

Bridge numbers for virtual and welded knots

Using Gauss diagrams, one can define the virtual bridge number ${\rm vb}(K)$ and the welded bridge number ${\rm wb}(K),$ invariants of virtual and welded knots with ${\rm wb}(K) \leq {\rm vb}(K).$ If $K$ is a classical knot, Chernov and Manturov showed that ${\rm vb}(K) = {\rm br}(K),$ the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group $G_K$, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group $\overline{G}_K$, and we apply these methods to address Questions 6.1, 6.2, 6.3 and 6.5 raised by Hirasawa, Kamada and Kamada in their paper "Bridge presentation of virtual knots," J. Knot Theory Ramifications 20 (2011), no. 6, 881--893.Comment: 15 pages, 9 figure

Sentiment d'efficacité personnelle et réussite scolaire au collégial

Le sentiment d’efficacité personnelle (Bandura), souvent confondu avec l’estime de soi, compte assurément au nombre des ingrédients nécessaires à la réussite des étudiants. La professeure-chercheuse à l’Université du Québec à Trois-Rivières, se propose de nous familiariser avec ce concept et met en relief les sources pouvant stimuler ce sentiment. Parmi ces sources, on retrouve les expériences de maitrise (découlant des efforts et des aptitudes), les expériences vicariantes (où les pairs sont pris en exemple), la persuasion verbale (procédant de l’influence des pairs) et les états physiologiques et émotionnels (touchant la dimension socioaffective)