Elliptic operators in odd subspaces

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of odd order. Index formulas are obtained for operators in odd subspaces on closed manifolds and for general boundary value problems. We prove that the eta-invariant of operators of odd order on even-dimesional manifolds is a dyadic rational number.Comment: 27 page

Heat Conduction in One-Dimensional chain of Hard Discs with Substrate Potential

Heat conduction of one-dimensional chain of equivalent rigid particles in the field of external on-site potential is considered. Zero diameters of the particles correspond to exactly integrable case with divergent heat conduction coefficient. By means of simple analytical model it is demonstrated that for any nonzero particle size the integrability is violated and the heat conduction coefficient converges. The result of the analytical computation is verified by means of numerical simulation in a plausible diapason of parameters and good agreement is observedComment: 14 pages, 7 figure

Intervalley-Scattering Induced Electron-Phonon Energy Relaxation in Many-Valley Semiconductors at Low Temperatures

We report on the effect of elastic intervalley scattering on the energy transport between electrons and phonons in many-valley semiconductors. We derive a general expression for the electron-phonon energy flow rate at the limit where elastic intervalley scattering dominates over diffusion. Electron heating experiments on heavily doped n-type Si samples with electron concentration in the range $3.5-16.0\times 10^{25}$ m$^{-3}$ are performed at sub-1 K temperatures. We find a good agreement between the theory and the experiment.Comment: v2: Notations changed: $\Delta_i$ --> $\delta v_i$, $\tau_{eff}$ removed. Eq. (1) changed, Eq. (2) added and complete derivation of Eq. (3) included. Some further discussion about single vs. many valley added [3rd paragraph after Eq. (7)]. End notes removed and new reference added [Kragler and Thomas]. Typos in references correcte

Adolescent girls’ osteoporosis knowledge and understanding with analysis of their current lifestyle choices

Adolescent girls especially, need to be aware of the possible onset of osteoporosis in future years so that it’s effects can at least be reduced if not totally prevented. Recently, it has been reported that one out of three adolescent girls had neither heard nor read anything about osteoporosis (Gurney, 2007). The aim of this research was to assess osteoporosis awareness and related behaviors amongst adolescent teenage girls with a view to help prevent osteoporosis occurring in later life. One hundred and forty two female students from a secondary comprehensive school aged 12-16 took part in a questionnaire. The questionnaire consisted of 40 questions and was split into four sections: Section A: An assessment of current osteoporosis knowledge. Section B: An assessment of dietary and physical activity knowledge. Section C: An assessment of current dietary intake for participant. Section D: An assessment of current physical activity for participant. General osteoporosis knowledge, risk factors, prevention, lifestyle behaviors, calcium intake, physical activity, smoking and alcohol intake were central aspects within the questionnaire. The following hypothesis was tested: that adolescent girls have adequate osteoporosis, dietary and physical activity knowledge and understanding (Hi). The null hypothesis (Ho) for this research was therefore given that adolescent girls have inadequate osteoporosis, dietary and physical activity knowledge and understanding. Descriptive statistics, using excel were used to tabulate and analyse the data. Section A focused on osteoporosis knowledge. An average of 31% of questions were answered correctly (mean score of 7 out of 23 correct answers and a standard deviation of 4.4). Section B focused on dietary and physical activity knowledge. An average of 57% of questions were answered correctly (mean score of 10 out of 18 correct answers and a standard deviation of 2.3). Section C focused on an assessment of current dietary intake, most calcium consumption amongst the adolescent girls occurred around one to three times a week which consisted on average of seven to twenty one calcium containing products. The results from section C also showed other products such as cakes, buns, biscuits, pizza, cola and fizzy drinks were consumed amongst adolescent girls most frequently two to three times a month. Overall the results from section C showed that most adolescent girls consume calcium rich products more frequently than the other less healthier portions of food. Section D focused on current physical activity levels, most of the adolescent girls spent two to four times a week participating in physical activity. Adolescent girls mostly tended to spend one to two hours per week and one two hours over the weekend watching the television (a relatively small proportion of their time). One to two hours was spent on homework during the week. Over the weekend one to two hours was also spent on homework. The null hypothesis (Ho) that adolescent girls have inadequate knowledge and understanding of osteoporosis was confirmed as the adolescent girls’ current knowledge of osteoporosis was found to be ‘poor’. Their dietary and physical activity knowledge however, was found to be ‘good’. Overall this research supports the null hypothesis. The average percentage of correct answers in sections A and B was 44% equating to ‘poor’ osteoporosis and related dietary and physical activity knowledge. Adolescent girls were generally unaware of osteoporosis issues and are currently doing moderate exercise and have a moderate consumption of calcium rich food, both of which must be increased to prevent the occurrence of osteoporosis in later life. Targeted education programmes are needed and should be aimed at improving osteoporosis knowledge to affect health beliefs and lifestyle choices in a manner appropriate and appealing to these girls. The National Osteoporosis Society is focusing on secondary education awareness in 2010

Elliptic operators in even subspaces

In the paper we consider the theory of elliptic operators acting in subspaces defined by pseudodifferential projections. This theory on closed manifolds is connected with the theory of boundary value problems for operators violating Atiyah-Bott condition. We prove an index formula for elliptic operators in subspaces defined by even projections on odd-dimensional manifolds and for boundary value problems, generalizing the classical result of Atiyah-Bott. Besides a topological contribution of Atiyah-Singer type, the index formulas contain an invariant of subspaces defined by even projections. This homotopy invariant can be expressed in terms of the eta-invariant. The results also shed new light on P.Gilkey's work on eta-invariants of even-order operators.Comment: 39 pages, 2 figure