Rectification of sets in Euclidean spaces

We criticize traditional definitions of the arc length which require semi-continuity from below. Symmetric definitions of lower and uppern-lengths (n-dimensional volumes) are introduced for a wide class of sets in Euclidean spaces, and the additivity of both functionals is proved.     

The umbral calculus and orthogonal polynomials

We determine all orthogonal polynomials having Boas-Buck generating functions g(t)(xf(t)), where% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHOo% qwcaGGOaGaamiDaiaacMcacqGH9aqpruqqYLwySbacfaGaa8hiamaa% BeaaleaacaaIWaaabeaakiaadAeacaqGGaWaaSbaaSqaaiaabgdaae% qaaOGaaeikaiaadggacaGGSaGaa8hiaiaadshacaqGPaGaaeilaiaa% bccacaqGGaGaaeiiaiaadggacqGHGjsUcaaIWaGaaiilaiaa-bcacq% GHsislcaaIXaGaaiilaiaa-bcacqGHsislcaaIYaGaaiilaiablAci% ljaacUdaaeaacqqHOoqwcaGGOaGaamiDaiaacMcacqGH9aqpcaWFGa% WaaSraaSqaaiaaicdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOm% aaqabaGccaGGOaWaaSqaaSqaaiaaigdaaeaacaaIZaaaaOGaaiilai% aa-bcadaWcbaWcbaGaaGOmaaqaaiaaiodaaaGccaGGSaGaa8hiaiaa% dshacaGGPaGaa8hiamaaBeaaleaacaaIWaaabeaakiaadAeacaqGGa% WaaSbaaSqaaiaabkdaaeqaaOGaaeikamaaleaaleaacaaIYaaabaGa% aG4maaaakiaacYcacaWFGaWaaSqaaSqaaiaaisdaaeaacaaIZaaaaO% Gaaiilaiaa-bcacaWG0bGaaiykaiaacYcacaWFGaWaaSraaSqaaiaa% icdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOmaaqabaGccaGGOa% WaaSqaaSqaaiaaisdaaeaacaaIZaaaaOGaaiilaiaa-bcadaWcbaWc% baGaaGynaaqaaiaaiodaaaGccaGGSaGaa8hiaiaadshacaGGPaGaai% 4oaaqaaiabfI6azjaacIcacaWG0bGaaiykaiabg2da9iaa-bcadaWg% baWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaacaqGZaaabe% aakiaacIcadaWcbaWcbaGaaGymaaqaaiaaisdaaaGccaGGSaGaa8hi% amaaleaaleaacaaIYaaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaS% qaaiaaiodaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGaaiykaiaa% -bcadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaaca% qGZaaabeaakiaabIcadaWcbaWcbaGaaGOmaaqaaiaaisdaaaGccaGG% SaGaa8hiamaaleaaleaacaaIZaaabaGaaGinaaaakiaacYcacaWFGa% WaaSqaaSqaaiaaiwdaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGa% aiykaiaacYcaaeaadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiam% aaBaaaleaacaqGZaaabeaakiaacIcadaWcbaWcbaGaaG4maaqaaiaa% isdaaaGccaGGSaGaa8hiamaaleaaleaacaaI1aaabaGaaGinaaaaki% aacYcacaWFGaWaaSqaaSqaaiaaiAdaaeaacaaI0aaaaOGaaiilaiaa% -bcacaWG0bGaaiykaiaacYcacaGGUaGaa8hiamaaBeaaleaacaaIWa% aabeaakiaadAeacaqGGaWaaSbaaSqaaiaabodaaeqaaOGaaeikamaa% leaaleaacaaI1aaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaSqaai% aaiAdaaeaacaaI0aaaaOGaaiilaiaa-bcadaWcbaWcbaGaaG4naaqa% aiaaisdaaaGccaGGSaGaa8hiaiaadshacaGGPaGaaiOlaaaaaa!C1F3!\[\begin{gathered}\Psi (t) = {}_0F{\text{ }}_{\text{1}} {\text{(}}a, t{\text{), }}a \ne 0, - 1, - 2, \ldots ; \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{2}} (\tfrac{1}{3}, \tfrac{2}{3}, t) {}_0F{\text{ }}_{\text{2}} {\text{(}}\tfrac{2}{3}, \tfrac{4}{3}, t), {}_0F{\text{ }}_{\text{2}} (\tfrac{4}{3}, \tfrac{5}{3}, t); \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{3}} (\tfrac{1}{4}, \tfrac{2}{4}, \tfrac{3}{4}, t) {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{2}{4}, \tfrac{3}{4}, \tfrac{5}{4}, t), \hfill \\{}_0F{\text{ }}_{\text{3}} (\tfrac{3}{4}, \tfrac{5}{4}, \tfrac{6}{4}, t),. {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{5}{4}, \tfrac{6}{4}, \tfrac{7}{4}, t). \hfill \\\end{gathered}\]We also determine all Sheffer polynomials which are orthogonal on the unit circle. The formula for the product of polynomials of the Boas-Buck type is obtained.     

Γ andX bandgap hydrostatic deformation potentials for epitaxial In0.52Al0.48As on InP(001)

The pressure dependence of the direct and indirect bandgap of epitaxial In_0.52Al_0.48As on InP(001) substrate has been measured using photoluminescence up to 92 kbar hydrostatic pressure. The bandgap changes from Γ toX at an applied pressure of ∼ 43 kbar. Hydrostatic deformation potentials for both the Γ andX bandgaps are deduced, after correcting for the elastic constant (bulk modulus) mismatch between the epilayer and the substrate. For the epilayer we obtain and+(2.81±0.15)eV for the Γ andX bandgaps respectively. From the pressure dependence of the normalized Γ-bandgap photoluminescence intensity a Γ-X lifetime ratio, (τ_Γ/τ _X ), of 4.1×10⁻³ is deduced.     

Polyethene with pendant fullerene moieties

Polyethene with fullerene moieties pendant on short-chain branches was prepared by the catalytic copolymerization of ethene and a fullerene-containing vinylic comonomer, yielding polyethene copolymers containing up to 25 wt% of C60.     

Thermal Stability Of Folic Acid in the solid-state

This study attempts to identify the degradative process which folic acid undergoes in the solid-state under thermal stress. In order to facilitate the process, the various pieces of the chemical structure, namely, p-amino benzoic acid, pterin and glutamic acid as both its d- and l-isomers were investigated as separate entities. These structured solid-state pieces were then compared to the composite solid state folic acid degradative curves in order to identify the peaks seen and provide direction for the interpolation of the degradative mechanism. It was observed that none of the structural pieces could be superimposed as assumed earlier and hence an attempt was made to identify the decomposition products using various analytical techniques such as infrared spectroscopy, mass spectroscopy and X-ray diffraction which suggested that the glutamic acid fragment is lost first as evidenced by acid loss and amide enhancement in the IR spectra. The vitamin was ultimately degrading to carbon fragments and that further identification was not necessary. This revised version was published online in August 2006 with corrections to the Cover Date.