The Geometry of Self-dual 2-forms

We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly independent vector fields on the sphere $S^{2n-1}$. We provide a direct proof that for $n$ odd ${\cal S}_{2n}$ has only one-dimensional linear submanifolds. We exhibit $2^c-1$ dimensional subspaces in dimensions which are multiples of $2^c$, for $c=1,2,3$. In particular, we demonstrate that the seven dimensional linear subspaces of ${\cal S}_{8}$ also include among many other interesting classes of self-dual 2-forms, the self-dual 2-forms of Corrigan, Devchand, Fairlie and Nuyts and a representation of ${\cal C}l_7$ given by octonionic multiplication. We discuss the relation of the linear subspaces with the representations of Clifford algebras.Comment: Latex, 15 page

A search on Dirac equation

The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given.Comment: 9 page

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields F_{\mu \nu}. We derive a topological bound on {\bf R}^8, \int_{M} ( F,F )^2 \geq k \int_{M} p_1^2 where p_1 is the first Pontrjagin class of the SO(n) Yang-Mills bundle and k is a constant. Strongly self-dual Yang-Mills fields realise the lower bound

Assessment Of The Curriculum Of Turkish Language Teaching In The 2nd Grade Of Primary Education In Terms Of Critical Thinking Skills

AbstractThis study aims to assess Turkish language teaching program of the 2nd grade primary school in terms of critical thinking skills. The document examination method, a qualitative research method, was employed in the research. Giving information on the general structure, and content of Turkish language teaching program (6-7-8), gains there from were reviewed. These gains were classified under the titles of critical thinking skills. In the reviewed Turkish language teaching program (6-7-8), 158 gains among the total of 228 under the fields of “reading, listening/following, speaking, writing, learning, and grammar” were found to be intended for critical thinking skills. It was further ascertained that, these gains were intended more for “application” and “analyzing” skills than those of critical thinking

Examination of critical thinking levels of fifth grade students in primary education in terms of their success in the Turkish language course

AbstractThis study was conducted in order to determine the relation between the critical thinking levels of 5th grade students in primary education and their success in Turkish Language course. The study was conducted in line with the relational scanning model. Target population of the study comprised primary education 5th grade students studying in the province of Erzurum. And the sample group comprised 610 students selected by convenience sampling from the counties of Palandöken and Yakutiye. Critical Thinking Scale developed by Demir (2006) was used in the study. In data analysis, by means of SPSS 17.0 software package, frequencies, arithmetic mean, standard deviation levels and, for the correlation between critical thinking levels and Turkish Language course success, correlation analysis were used. In conclusion of the study, it was determined that the criticial thinking levels of students were high and there was a positive correlation between critical thinking levels and success in the Turkish Language course

Patterns of COVID-19-related headache: A cross-sectional study

Background: Headache is the most common COVID-19-related neurological symptom. We investigated the characteristics of COVID-19-related headache and their relationship with clinical severity in Kirs , ehir Province, Turkey. Methods: This cross-sectional study prospectively enrolled 226 COVID-19-positive patients who developed headache during acute infection. Demographic data, headache characteristics, and infection symptoms were recorded. The clinical severity of COVID-19 was documented in each participant. Result: New-onset COVID-19-related headaches lasting 4 days were reported in 164 patients (72.5 %); these were mostly bilaterally or localized to the forehead (58.4 %), pulsating (42.5 %), moderate to severe intensity (30.1 %), with a partial response to paracetamol (23.5 %). The other 62 patients (27.4 %) reported headaches before COVID-19. Their COVID-related headaches were fiery type (p = 0.025), of very severe intensity (p = 0.008), had a holocranial distribution (p = 0.004), and were less response to paracetamol (p = 0.003); the headaches were significantly more frequent after COVID-19 than before COVID-19. Older age, high body mass index, and low education level were significantly higher in the severe group (all p < 0.001). Female sex (p = 0.019) and being a healthcare worker (p < 0.001) were significantly more frequent in mild cases. Conclusions: Bilateral, prolonged, moderate to severe headaches that were analgesic resistant are more frequent in patients with COVID-19 infection. Further study should examine whether the headache characteristics distinguish COVID-19-related headaches from other types, particularly in asymptomatic subjects