Unequal Sides Research Articles

Epstein zeta function and toy p-branes from old string action

The authors study toy p-branes from old string action, and regularize their Casimir energies by the Epstein zeta function. They obtain the special dimensions for the open (p=2, 3, 4) and closed (p=2, 3, 4, 6, 8) branes with equal sides. The upper bound of p is obtained. They also discuss the special dimensions for open and closed membranes with unequal sides. The resulting dimensions may be integers for the case of unequal sides.

Correction of intensities for preferred orientation in powder diffractometry: application of the March model

Diffracted intensities from axially symmetric flat-plate or capillary specimens, composed of effectively rod- or disk-shaped crystallites, can be corrected for preferred orientation with a single pole-density profile. A convenient procedure is to approximate this profile with a function whose variable parameters are fit during least-squares structure refinement. Several functions have previously been suggested but without theoretical justification. The present study reviews the derivation of this method and examines its assumptions and applications. The several proposed functions are compared with each other and with the March function which describes the pole-density distribution produced by rigid-body rotation of inequant crystallites (i.e. crystallites with unequal sides) upon axially symmetric volume-conserving compression or expansion. For its basis, ease of use, single variable parameter, direct interpretability and good refinement test results, the March distribution is proposed as an advantageous pole-density profile function for general use.

Ancient Versions of two Trigonometric Lemmas

To justify certain steps of the computation developed in his Sand-Reckoner, Archimedes cites (without proof) the following inequalities relative to the sides of right triangles:if of two right-angled triangles, (one each of) the sides about the right angle are equal (to each other), while the other sides are unequal, the greater angle of those toward [sc. next to] the unequal sides has to the lesser (angle) a greater ratio than the greater line of those subtending the right angle to the lesser, but a lesser (ratio) than the greater line of those about the right angle to the lesser.That is, with reference to the two right triangles ABG, DEZ (Fig. 1), where AG equals DZ and the angle at B is greater than that at E, ZE:GB < angle B:angle E < DE:AB.

Kinetics and mechanism of glucose electrooxidation on different electrode-catalysts: Part I. Adsorption and oxidation on platinum

The processes of adsorption and electrooxidation of glucose on a smooth platinum electrode have been investigated in a wide range of pH values. It is found that glucose adsorption are platinum is accompanied by dehydrogenation of adsorbed molecules. The θ R vs. E r dependence represents a bell-shaped curve with unequal sides and with a maximum at E r = 0.2 V at 0 < pH < 12 or at E r = 0.4 when pH > 12. The kinetics of adsorption is described by the Roginsky-Zel'dovich equation, and the dependence of the steady-state coverage on the glucose bulk concentrations by the Temkin isotherm. It is shown that in the case of glucose adsorption on platinum Q dehyd. H > Q H, i.e. when glucose is brought into contact with a platinum electrode, the catalytic decomposition of glucose molecules occurs in addition to the formation of strongly chemisorbed particles. The transient current at E r < 1.0 V is a current due to the ionization of hydrogen formed during adsorption with dehydrogenation of glucose and its catalytic decomposition. The glucose electrooxidation rate under steady-state conditions at E r < 0.7 V is determined by the interaction of the chemisorbed carbon-containing particle with OH ads. The slow step of glucose electrooxidation in the potential range 1.0 < E r < 1.5 V is the interaction of glucose molecules from the solution bulk with the surface platinum oxide, the latter undergoing a quick electrochemical regeneration thereafter. The basic regularities and mechanism of glucose electrooxidation on platinum are shown to be analogous to those obtained earlier for such elementary organic fuels as formaldehyde and formic acid.

Synthesis, spectroscopic characterization and crystal and molecular structure of HRu 3(OCN(CH 3) 2)(CO) 10

Dimethylamine reacts with Ru 3(CO) 12 to produce the η 2-hydrido-η-formamido cluster complex HRu(OCN(CH 3) 2)(CO) 10 (I). This formulation is consistent with spectroscopic features such as the absence of v(NH) in the infrared, the presence in the Raman of v(RuHRu) at 1400 cm −1 ( v(RuDRu) at 990 cm −1) and indication in the 1H NMR of diastereotopic methyl groups bonded to the nitrogen atom. Since these data could not lead to an unequivocal structure assignment a single crystal X-ray study at 115 K was undertaken. The complex crystallizes in the triclinic space group, P 1 with cell dimensions; a 7.299(33) », b 9.5037(40) », c 13.7454(57) », α 91.876(34)°, β 96.387(34)°, γ 95.341(34)° and Z = 2. The structure was solved by a combination of Patterson and Fourier techniques and refined by full matrix least squares to a final R = 0.054 and R ω = 0.074 for 3074 unique reflections. The three ruthenium atoms define a triangle of unequal sides with both the hydride and formamido groups bridging the longest edge; the formamido group is coordinated through the carbon and oxygen atoms. The edge of the ruthenium triangle bridged both by the hydrogen atom and the formamido group is 2.8755(15) »; the other two edges of the ruthenium triangle are observed to be 2.8319(15) and 2.8577(14) », respectively. In the formamido group the distance CO 1.287(9) » and CN 1.340(10) » reflect partial double bond charater in each bond consistent with observation of two chemically distinct methyl groups on the dinitrogen atom. The hydrogen atom bridging one edge of the ruthenium triangle is asymmetrically positioned at 1.73(9) » from the ruthenium atom bonded to the oxygen atom and 1.91(9) » from the ruthenium atom bonded to the carbon atom of the carboxamido group.