A variant of Lehmer's conjecture

Lehmer's conjecture asserts that τ(p)≠0 where τ is the Ramanujan τ-function. This is equivalent to the assertion that τ(n)≠0 for any n. A related problem is to find the distribution of primes p for which . These are open problems. We show that the variant of estimating the number of integers n for which n and τ(n) do not have a non-trivial common factor is more amenable to study. In particular, we show that the number of such n?x is ?x/logloglogx. We prove a similar result for more general cusp forms. This may be seen as a modular analogue of an old result of Erd?s on the Euler ? function.     

Regioselective strategies mediated by lanthanide triflates for efficient assembly of oligomannans

Readily prepared mannosyl n-pentenylorthoesters (NPOEs) serve as donors in themselves and as convenient intermediates for other glycosyl donors, such as n-pentenyl glycosides (NPGs), thioglycosides, and trichloroacetimidates. These various donors are activated by different reagents, and are therefore amenable to versatile, discriminate use. Scandium and ytterbium triflates respond very differently to these donors, with the result that chemoselective discrimination between NPOEs, NPGs, trichloroacetimidates as well as ethyl and phenyl thioglycosides can be achieved. Appropriate NPOEs are also able to provide 2,6 and 3,6 diol acceptors via rearrangement or glycoside formation, and these can be used for one-pot, sequential glycosidations based on orthogonal donors, and in situ double differential glycosidations. Thus NPOEs activated by iodonium ion, specifically generated from ytterbium triflate/N-iodosuccinimide, can be used to monoglycosidate the diols rapidly, with exquisite regio, and sometimes chemo, selectivity. The residual NPOE is converted into disarmed NPG, which is refractory to the reaction conditions, and so poses no threat to the free-OH of the monoglycosidation product. Further glycosidation of the latter can then achieved by direct addition of a trichloroacetimidate or ethyl thioglycoside. This basic strategy has been used to prepare a branched chain pentadecamannan. The success is an example of the efficiency of donor/acceptor MATCH concept for regioselective glycosylation.     

Thermodynamic prediction of bulk metallic glass forming alloys in ternary Zr–Cu–X (X = Ag,Al, Ti,Ga) systems

Prediction of bulk metallic glass (BMG) forming compositions has always been a challenge due to thermodynamic and kinetic constraints. In the present investigation, a parameter based on the enthalpy of chemical mixing (?H_chem) and the mismatch entropy (?S_σ/k_B) has been used to correlate with glass forming ability in some Zr based BMGs. The new thermodynamic parameter, P_HS = ?H_chem × ?S_σ/k_B, is found to have strong correlation with glass forming ability in the configurational entropy (?S_config/R) range of 0.9–1.0. P_HS has been calculated for compositions in Zr–Cu–Ag, Zr–Cu–Al, Zr–Cu–Ti and Zr–Cu–Ga ternary systems. It is observed that in all the systems studied, the best BMG composition (highest critical diameter (Z_c) of glass formation) is the one that corresponds to the highest negative P_HS value. Present approach using P_HS could be road map to design new BMG forming compositions.     

A problem in enumerating extreme points, and an efficient algorithm for one class of polytopes

We consider the problem of developing an efficient algorithm for enumerating the extreme points of a convex polytope specified by linear constraints. Murty and Chung (Math Program 70:27–45, 1995) introduced the concept of a segment of a polytope, and used it to develop some steps for carrying out the enumeration efficiently until the convex hull of the set of known extreme points becomes a segment. That effort stops with a segment, other steps outlined in Murty and Chung (Math Program 70:27–45, 1995) for carrying out the enumeration after reaching a segment, or for checking whether the segment is equal to the original polytope, do not constitute an efficient algorithm. Here we describe the central problem in carrying out the enumeration efficiently after reaching a segment. We then discuss two procedures for enumerating extreme points, the mukkadvayam checking procedure, and the nearest point procedure. We divide polytopes into two classes: Class 1 polytopes have at least one extreme point satisfying the property that there is a hyperplane H through that extreme point such that every facet of the polytope incident at that extreme point has relative interior point intersections with both sides of H; Class 2 polytopes have the property that every hyperplane through any extreme point has at least one facet incident at that extreme point completely contained on one of its sides. We then prove that the procedures developed solve the problem efficiently when the polytope belongs to Class 2.     

Electrospray ionisation mass spectral studies on hydrolysed products of sulfur mustards

Off-site detection of the hydrolysed products of sulfur mustards in aqueous samples is an important task in the verification of Chemical Weapons Convention (CWC)-related chemicals. The hydrolysed products of sulfur mustards are studied under positive and negative electrospray ionisation (ESI) conditions using an additive with a view to detecting them at trace levels. In the presence of cations (Li(+), Na(+), K(+) and NH(4) (+)), the positive ion ESI mass spectra of all the compounds include the corresponding cationised species; however, only the [M+NH(4)](+) ions form [M+H](+) ions upon decomposition. The tandem mass (MS/MS) spectra of [M+H](+) ions from all the hydrolysed products of the sulfur mustard homologues were distinct and allowed these compounds to be characterised unambiguously. Similarly, the negative ion ESI mass spectra of all the compounds show prominent adducts with added anions (F(-), Cl(-), Br(-), and I(-)), but the [M-H](-) ion can only be generated by decomposition of an [M+F](-) ion. The MS/MS spectra of the [M-H](-) ions from all the compounds result in a common product ion at m/z 77. A precursor ion scan of m/z 77 is shown to be useful in the rapid screening of these compounds in aqueous samples at trace levels, even in the presence of complex masking agents, without the use of time-consuming sample preparation and chromatography steps. An MS/MS method developed to measure the detection limits of the hydrolysed products of sulfur mustards found these to be in the range of 10-500 ppb.     

Maxwell–Wagner polarization in grain boundary segregated NiCuZn ferrite

The present work investigates the polarization response in polycrystalline Ni_0.9−yCu_yZn_0.1Fe_1.98O_4−δ (y = 0, 0.1, 0.2, 0.3, 0.4, 0.5) ferrite synthesized by solid–state reaction method. X-ray diffraction analysis confirmed cubic spinel phase formation in the calcined samples. Sintered samples contain a continuous network of CuO-rich segregation along the grain boundaries for y ≥ 0.2. Dielectric spectra showed a relaxation peak for y ≥ 0.2 in the frequency range of 1 kHz–1 MHz. This relaxation has been explained based on Maxwell–Wagner polarization considering two-layer model in connection with two heterogeneous dielectric media.     

Photoelectric effect in some alloys in the region of absorption edges

Summary Photoelectric effect in alloys using the transmission method in the energy range (17.8÷138) keV, at energies in the region of absorption edges of the various elements in the alloys, have been measured and compared with the respective theoretical estimates.     

The Army Training Mix Model

We describe a model for making decisions concerning the proper mix of TADSS (training aids, devices, simulators and simulations) and conventional training resources, and the best modes of using them to maintain soldier and unit proficiency. Given the proficiency standards, the model determines the resources needed, the training methods for using them, and the frequency with which each method needs to be repeated, in order to maintain the standards at minimum cost (sum of equipment procurement costs and operational costs). The model is illustrated with an example problem dealing with the analysis of training systems in a battalion training strategy. Our model has much wider applicability: it can be used for evaluating and determining minimum cost training strategies for other combined arms elements, higher level units and other training scenarios under a variety of circumstances.