On the maximal difference between an element and its inverse in residue rings

We investigate the distribution of where

Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on . We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study than a more traditional way using exponential sums.

     

The geology of the manganese ore deposits of talwara,district Banswara,Rajasthan

Talwara (23° 34′: 74° 22′) is a small Manganese-producing locality in the district of Banswara, Rajasthan. From the Banswara Town, Talwara can be reached by a motorable road. In the past, the Manganese ores of the locality have drawn very little attention of the geologists. The ores of Manganese, being associated exclusively with the Aravalli limestone of Rajasthan, occur in the form of scattered masses occupying solution cavities, fissures, etc., in limestone. Aravalli quartzites, conglomerates, phyllites and slates also occur in the vicinity of Talwara. The white limestone with a mosaic structure is of common occurrence and is composed of calcite and dolomite with some magnetite. The impurer varieties of limestone are characterised by the presence of tremolite, hornblende and quartz with a little of biotite. Schistosity is developed only in a calc-amphibole rock having alternating bands of hornblende and limestone. Crush-breccia is noticed occurring in association with manganiferous limestone at a distance of about 2 miles west of Talwara. This breccia might have originated due to severe local crushing of the country-rock. The limestones seemed to be a product of low-grade metamorphism of a calcareous sediment. Fresh ores are generally hard and cavernous but on weathering they become softer and soily. Fragments of unreplaced limestone are occasionally present in the ore-bodies. This limestone has also been replaced by the Manganese ores to a greater or lesser extent, depending on the structural characters of the host rock. The ores are composed largely of massive and colloform cryptomelane. Pyrolusite forms an insignificant part of the ores. There are two generations of cryptomelane, of which, the earlier colloform one is veined by the later generation of cryptomelane. There are also examples of pseudomorphous replacement of limestone by cryptomelane. Most of the pyrolusite was derived from the supergene alteration of cryptomelane. From their nature and mode of occurrence the Manganese ores are classified as outcrop secondary ores formed in cavities in crystalline limestones belonging to the less metamorphosed type of Dharwar.     

The direct electrochemical synthesis and crystal structure of salts of [M4(SC6H5)10]2− anions (M = Zn,Cd)

The salts [(C₂H₅)₃NH]₂[M₄(SC₆H₅)₁₀] (M = Zn, Cd) can be prepared by the electrochemical oxidation of the metal in an acetonitrile solution of triethylamine and benzenethiol. An X-ray crystal structure determination shows that the anion consists of a tetrahedron of metal atoms, each carrying a terminal -SC₆H₅ ligand, and connected to the three other metal atoms by bridging > SC₆H₅ groups. The results are compared with those for similar compounds reported in the literature.     

The reflection spectrum of cuprous chloride in the 4.5−30 eV range and the related band structure

The reflection spectrum of CuCl is measured on cleaved monocrystals, at low temperature, up to 30 eV. The main lines of different band schemes, proposed recently by theoretical approaches, are compared with characteristics features in the reflection spectra. All transitions at ?, between the three highest valence bands and the two lowest conduction bands are identified. The observation of transitions where the initial states are in 3sCl^- permits to determine critical points in the conduction bands. With the help of our experimental data we propose a band scheme for cuprous chloride.     

Approaches to open fullerenes: a 1,2,3,4,5,6-hexaadduct of C60

Complete saturation of a single six-membered ring on fullerene C60 has been achieved. The critical step in this first synthesis of a fully characterized 1,2,3,4,5,6-hexaadduct consisted of a remarkable double 5-exo-trig addition of alkoxyl radicals promoted by lead tetraacetate. Two possible opening pathways ([2 + 2 + 2] retrocycloadditions) for the newly synthesized compound were explored using quantum mechanical calculations. We found that the oxa bridges in the hexaadduct prevent ring opening through the retro[2 + 2 + 2] mechanism due to the high activation barrier and endothermicity of the reaction.     

Classification of Teleparallel Homothetic Vector Fields in Cylindrically Symmetric Static Space-Times in Teleparallel Theory of Gravitation

In this paper we classify cylindrically symmetric static space-times according to their teleparallel homothetic vector fields using direct integration technique. It turns out that the dimensions of the teleparallel homothetic vector fields are 4, 5, 7 or 11, which are the same in numbers as in general relativity. In case of 4, 5 or 7 proper teleparallel homothetic vector fields exist for the special choice to the space-times. In the case of 11 teleparallel homothetic vector fields the space-time becomes Minkowski with all the zero torsion components. Teleparallel homothetic vector fields in this case are exactly the same as in general relativity. It is important to note that this classification also covers the plane symmetric static space-times.     

Multiscaling analysis of a slowly varying anaerobic digestion model

We construct the slowly varying limiting state solutions to a nonlinear dynamical system for anaerobic digestion with Monod-based kinetics involving slowly varying model parameters arising from slow environmental variation. The advantage of these approximate solutions over numerical solutions is their applicability to a wide range of parameter values. We use these limiting state solutions to develop analytic approximations to the full nonlinear system by applying a multiscaling technique. The approximate solutions are shown to compare favorably with numerical solutions.     

A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces

We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.     

Switch of electronic reactivity in fullerene C60: activation of three trans-4 positions via temporary saturation of the cis-1 positions

[Structure: see text] The regioselective functionalization of C60 with a trans-4,trans-4,trans-4 trisaddition pattern is not feasible directly. We have found an indirect approach taking advantage of the modified electronic reactivity of cis-1 bisadducts. The cis-1 addition pattern electronically activates three trans-4 C=C bonds on the opposite hemisphere of C60, allowing further highly regioselective additions at these positions. Thermal removal of the cis-1 blocking unit results in a trans-4,trans-4,trans-4 trisadduct with C3v symmetry.