On the commutativity degree of compact groups

In any finite group G, the commutativity degree of G (denoted by d(G)) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the nth commutativity degree (denoted by d _n (G)) is the probability that a randomly chosen ordered (n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d(G) and d _n (G) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).     

Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces

Let (X,?) be a partially ordered set and d be a complete metric on X. Let F,G be two set-valued mappings on X. We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X.     

Surface-directed boundary flow in microfluidic channels

Channel geometry combined with surface chemistry enables a stable liquid boundary flow to be attained along the surfaces of a 12 microm diameter hydrophilic glass fiber in a closed semi-elliptical channel. Surface free energies and triangular corners formed by PDMS/glass fiber or OTS/glass fiber surfaces are shown to be responsible for the experimentally observed wetting phenomena and formation of liquid boundary layers that are 20-50 microm wide and 12 microm high. Viewing this stream through a 20 microm slit results in a virtual optical window with a 5 pL liquid volume suitable for cell counting and pathogen detection. The geometry that leads to the boundary layer is a closed channel that forms triangular corners where glass fiber and the OTS coated glass slide or PDMS touch. The contact angles and surfaces direct positioning of the fluid next to the fiber. Preferential wetting of corner regions initiates the boundary flow, while the elliptical cross-section of the channel stabilizes the microfluidic flow. The Young-Laplace equation, solved using fluid dynamic simulation software, shows contact angles that exceed 105 degrees will direct the aqueous fluid to a boundary layer next to a hydrophilic fiber with a contact angle of 5 degrees. We believe this is the first time that an explanation has been offered for the case of a boundary layer formation in a closed channel directed by a triangular geometry with two hydrophobic wetting edges adjacent to a hydrophilic surface.     

Synthesis of spongy gold nanocrystals with pronounced catalytic activities

A simple and easier chemical method for preparing spongy gold nanocrystals has been developed on the basis of a modified-citrate reduction technique of the corresponding gold salt at 25 degrees C in the absence of template. These nanocrystals possessed autocatalytic behavior and exhibited pronounced catalytic activity in the borohydride reduction of 4-nitrophenol due to their unique spongy morphology.     

Strong resonance enhancement of a single harmonic generated in the extreme ultraviolet range

We report the demonstration of strong resonance enhancement of a single high-order harmonic in the extreme ultraviolet (XUV) region generated from the interaction of a femtosecond pulse with low-ionized In ablation. A strong 13th harmonic (61.2 nm) of Ti:sapphire laser radiation with output intensity almost two orders of magnitude higher than neighboring harmonics was observed in these studies. The high conversion efficiency of the 13th harmonic (8 x 10(-5)) is attributed to multiple collisions of electron trajectories with the origin due to multiphoton resonance with the In ion.     

The radiation shielding offered by the commercial glass installed in Bangladeshi dwellings

One sign of a vibrant Bangladeshi economy has been the move away from the use of more traditional housing materials towards a preference for modern constructional media. Glass, one such example, used both decoratively and in a structural context, offers various advantageous properties and facets including a protective feature against radiation that has not previously been considered. Current interest examines the dosimetric possibilities offered by the commercial glass as a secondary shield and also in retrospective ionising radiation exposure analysis. Four popular brands of window glass are investigated, all available within the local market (PHP-Bangladesh, Usmania-Bangladesh, Nasir-Bangladesh and Rider-China), all with the same thickness and colour, varying in terms of elemental weight fractions as evaluated by energy dispersive X-ray analysis. As potential attenuators of transmitted radiation thereby forming secondary barriers against radiation exposure from penetrating radiations, the four brands of glass have been studied using photon energies from 59 up to 1332 keV, a range of values representative of that potentially encountered in incidents. Use has been made of a well-shielded high-purity germanium γ-ray spectrometer and associated electronics, providing for evaluation of the characteristic barrier parameters of half-value layer, radiation protection efficiency and effective atomic number (Z_eff). Of the four brands investigated, Rider provides superior secondary shielding performance. Concerning potential retrospective dosimetry the effective atomic number of the glass samples are comparable with that of the commercial thermoluminescence (TL) dosimeter TLD-200. At high doses, the TL yields are sufficient to provide for retrospective accident dosimetry.     

Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method

In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed graph.We establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of f.Indeed,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular.An application of this method to variational inequality is given.In addition,a numerical experiment is given which illustrates the theoretical result.     

The regularity of edge ideals of graphs

In this paper, we introduce some reduction processes on graphs which preserve the regularity of related edge ideals. Using these, an alternative proof for the theorem of R. Fröberg on the linearity of the resolutions of edge ideals is given.     

Modeling the interfacial effect on the yield strength and flow stress of thin metal films on substrates

It is shown in this paper that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening rates (flow stress) of metallic thin films on elastic substrates. This is achieved by developing a higher-order strain gradient plasticity theory based on the principle of virtual power and the laws of thermodynamics. This theory enforces microscopic boundary conditions at interfaces which relate a microtraction stress to the interfacial energy at the interface. It is shown that the film bulk length scale controls the size effect if a rigid interface is assumed whereas the interfacial length scale dominates if a compliant interface is assumed.