Diophantine approximation with multiplicative functions

Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that $$\left|\frac{\sigma(n)}{n}-\alpha \right| < n^{-0.52}.$$ We prove this result by modifying an argument given by Wolke (Monatsh Math 83:163–166, 1977) which in its original form could not produce an exponent greater than 0.5. We also explain how the exponent can be improved to 0.61 on the Riemann Hypothesis.     

Metrical Theorems on Restricted Diophantine Approximations to Points on a Curve

We investigate rational approximations (r/p,q/p) or (r/p,q/r) to points on the curve (α,α^τ) for almost all α>0, where p,q,r are all primes. We immediately obtain corollaries on making p,[αp], [α^τp] simultaneously prime.     

Inner-shell excitation spectroscopy and X-ray photoemission electron microscopy of adhesion promoters

The C 1s, Si 2p, Si 2s, and O 1s inner-shell excitation spectra of vinyltriethoxysilane, trimethylethoxysilane, and vinyltriacetoxysilane have been recorded by electron energy loss spectroscopy under scattering conditions dominated by electric dipole transitions. The spectra are converted to absolute optical oscillator strength scales and interpreted with the aid of ab initio calculations of the inner-shell excitation spectra of model compounds. Electron energy loss spectra recorded in a transmission electron microscope on partly cured adhesion promoter, atomic force micrographs, and images and X-ray absorption spectra from X-ray photoemission electron microscopy of as-spun and cured vinyltriacetoxysilane-based adhesion promoter films on silicon are presented. The use of these measurements in assisting chemistry studies of adhesion promoters for electronics applications is discussed.     

PRIMARY PROCESSES IN THE PHOTOCHEMISTRY OF AQUEOUS SULPHACETAMIDE: A LASER FLASH PHOTOLYSIS and PULSE RADIOLYSIS STUDY

Abstract— The spectra have been measured of the transient species formed in the nanosecond flash photolysis of aqueous solutions of sulphacetamide under a variety of conditions. In addition to the excited triplet state, the cation radical and the solvated electron were observed. The ionisation of aqueous sulphacetamide was found to occur by a biphotonic process. The extinction coefficient of the cation radical of sulphacetamide was determined by both laser flash photolysis and pulse radiolysis techniques, a value of 1.9 times 10₃ dm₃mol_-1cm_-1 being obtained. The rate of electron reaction with sulphacetamide and the anion radical spectrum were also determined by the two techniques, good agreement being obtained. The spectrum of the product of the reaction of the superoxide anion radical and the corresponding rate constant have also been determined. A possible mechanism of photosensitized skin reaction due to sulphacetamide is discussed.     

PRIMARY PROCESSES IN THE LASER FLASH PHOTOLYSIS AND PULSE RADIOLYSIS OF l-EPHEDRINE

Abstract— Transient absorption spectra produced by laser flash-photolysis of an aqueous solution of ephedrine have been measured under a variety of conditions. Ephedrine was found to photoionise via a biphotonic process. The apparent yield of photoionisation increases with lowering of pH, a value of 8.8 being found for the p K _a associated with this change. The cation radical absorption spectrum has been determined using the techniques of both pulse radiolysis and laser flash photolysis. The extinction coefficient of the cation at 295 nm was determined to be 1.37 × 10⁴ dm³ mol^-1 cm^-1 and 1.2 × 10⁴ dm³ mol-¹cm-¹ by the two techniques, respectively, at pH 11. It is also shown that the rate constant for electron abstraction by the azide radical to form the ephedrine cation is controlled by protonation of the amine group in the side chain. The ephedrine anion radical spectrum and its extinction coefficient at 305 nm were also determined. The excited states responsible for photoionisation and photodegradation are discussed.     

Kinetics of nucleophilic attack on coordinated organic moieties : VI. Mechanism of addition of tri-n-butylphosphine to [(C7H7)M(CO)3]BF4 (M = Cr, Mo, W) and related cations

Phosphonium adduct formation via attack of tri-n-butylphosphine on the cations [(C₇H₇)M(CO)₃]⁺ (M = Cr, Mo, W) obeys the rate law, Rate = k [complex] [PBu₃]. The very similar rate constants for the Cr, Mo and W complexes confirm the similar electrophilicities of the tropylium rings in these cations, and also support the view that there is direct addition to the rings. The related complexes [(C₆H₇)Fe(CO)₃]BF₄ and [(C₆H₆)Mn(CO)₃]BF₄ also form adducts with PBu₃, and the quantitative reactivity order [(C₆H₇)Fe(CO)₃]⁺ > [(C₇H₇)Cr(CO)₃]⁺ » [(C₆H₆)Mn(CO)₃]⁺ (160:60:1) has been established.     

Conventional MR imaging with simultaneous measurements of cerebral blood volume and vascular permeability in ganglioglioma

The conventional MR imaging appearance of gangliogliomas is often variable and nonspecific. Conventional MR images, relative cerebral blood volume (rCBV) and vascular permeability (K(trans)) measurements were reviewed in 20 patients with pathologically proven grade 1 and 2 gangliogliomas (n = 20) and compared to a group of grade 2 low-grade gliomas (n = 30). The conventional MRI findings demonstrated an average lesion size of 4.1 cm, contrast enhancement (n = 19), variable degree of edema, variable mass effect, necrosis/cystic areas (n = 8), well defined (n = 12), signal heterogeneity (n = 9), calcification (n = 4). The mean rCBV was 3.66 +/- 2.20 (mean +/- std) for grade 1 and 2 gangliogliomas. The mean rCBV in a comparative group of low-grade gliomas (n = 30), was 2.14 +/- 1.67. p Value < 0.05 compared with grade 1 and 2 ganglioglioma. The mean K(trans) was 0.0018 +/- 0.0035. The mean K(trans) in a comparative group of low-grade gliomas (n = 30), was 0.0005 +/- 0.001. p Value = 0.14 compared with grade 1 and 2 ganglioglioma. The rCBV measurements of grade 1 and 2 gangliogliomas are elevated compared with other low-grade gliomas. The K(trans), however, did not demonstrate a significant difference. Gangliogliomas demonstrate higher cerebral blood volume compared with other low-grade gliomas, but the degree of vascular permeability in gangliogliomas is similar to other low-grade gliomas. Higher cerebral blood volume measurements can help differentiate gangliogliomas from other low-grade gliomas.     

Metastability of native proteins and the phenomenon of amyloid formation

An experimental determination of the thermodynamic stabilities of a series of amyloid fibrils reveals that this structural form is likely to be the most stable one that protein molecules can adopt even under physiological conditions. This result challenges the conventional assumption that functional forms of proteins correspond to the global minima in their free energy surfaces and suggests that living systems are conformationally as well as chemically metastable.     

Spinning solutions of classical Yang-Mills theory

A class of 0(4)-covariant solutions is found to Yang-Mills field equations in Minkowski space. The solutions have finite energy, contain the 0(4)-invariant solutions including the meron as special cases, and have an unexpected new feature: they possess angular momentum. Their physical properties are examined and numerical solutions are displayed graphically.     

UV, VUV and soft X-ray photoabsorption of dimethyl ether by dipole (e,e) spectroscopies

Absolute UV and VUV photoabsorption oscillator strengths (cross-sections) for the valence shell discrete and continuum regions of dimethyl ether (CH₃OCH₃, DME) have been measured from 5 to 32 eV using high resolution (HR) (0.05 eV f.w.h.m.) dipole (e,e) spectroscopy. A wide-range spectrum, spanning the UV, VUV and soft X-ray regions, from 5 to 200 eV has also been obtained at low resolution (LR) (1 eV f.w.h.m.), and this has been used to determine the absolute oscillator strength scale by employing valence shell Thomas–Reiche–Kuhn (i.e., S(0)) sum-rule normalization. The presently reported HR and LR absolute photoabsorption oscillator strengths are compared with previously published data from direct photoabsorption measurements in those limited energy regions where such data are available. Evaluation of the S(−2) sum using the presently reported absolute differential photoabsorption oscillator strength data gives a static dipole polarizability for dimethyl ether in excellent agreement (within 0.5%) with previously reported polarizability values. Other dipole sums S(u), (u=−1,−3,−4,−5,−6,−8,−10), and logarithmic dipole sums L(u), (u=−1 to −6), are also determined from the presently reported absolute differential photoabsorption oscillator strength data using dipole sum rules.