Chebyshev centers and fixed point theorems

Brodskii and Milman proved that there is a point in C(K)$C(K)$, the set of all Chebyshev centers of K, which is fixed by every surjective isometry from K into K whenever K   is a nonempty weakly compact convex subset having normal structure in a Banach space. Motivated by this result, Lim et al. raised the following question namely “does there exist a point in C(K)$C(K)$ which is fixed by every isometry from K into K?”. In fact, Lim et al. proved that “if K is a nonempty weakly compact convex subset of a uniformly convex Banach space, then the Chebyshev center of K is fixed by every isometry T from K into K”. In this paper, we prove that if K   is a nonempty weakly compact convex set having normal structure in a strictly convex Banach space and F$\mathfrak{F}$ is a commuting family of isometry mappings on K   then there exists a point in C(K)$C(K)$ which is fixed by every mapping in F$\mathfrak{F}$.     

Propylphosphonic anhydride (T3P): an efficient reagent for the one-pot synthesis of 1,2,4-oxadiazoles, 1,3,4-oxadiazoles, and 1,3,4-thiadiazoles

Propylphosphonic anhydride (T3P^®) has been demonstrated to be an efficient and mild reagent for the one-pot synthesis of 1,2,4-oxadiazoles, 1,3,4-oxadiazoles, and 1,3,4-thiadiazoles from carboxylic acids.     

Direct electrochemistry of glucose oxidase and sensing glucose using a screen-printed carbon electrode modified with graphite nanosheets and zinc oxide nanoparticles

We have studied the direct electrochemistry of glucose oxidase (GOx) immobilized on electrochemically fabricated graphite nanosheets (GNs) and zinc oxide nanoparticles (ZnO) that were deposited on a screen printed carbon electrode (SPCE). The GNs/ZnO composite was characterized by using scanning electron microscopy and elemental analysis. The GOx immobilized on the modified electrode shows a well-defined redox couple at a formal potential of −0.4 V. The enhanced direct electrochemistry of GOx (compared to electrodes without ZnO or without GNs) indicates a fast electron transfer at this kind of electrode, with a heterogeneous electron transfer rate constant (K_s) of 3.75 s⁻¹. The fast electron transfer is attributed to the high conductivity and large edge plane defects of GNs and good conductivity of ZnO-NPs. The modified electrode displays a linear response to glucose in concentrations from 0.3 to 4.5 mM, and the sensitivity is 30.07 μA mM⁻¹ cm⁻². The sensor exhibits a high selectivity, good repeatability and reproducibility, and long term stability.

Graphical representation for the fabrication of GNs/ZnO composite modified SPCE and the immobilization of GOx

     

Vickrey–Dutch procurement auction for multiple items

We consider a setting where there is a manufacturer who wants to procure multiple items from a set of suppliers each of whom can supply one or more of these items (bundles). We design an ascending price auction for such a setting which implements the Vickrey–Clarke–Groves outcome and truthful bidding is an ex post Nash equilibrium. Our auction maintains non-linear and non-anonymous prices throughout the auction. This auction has a simple price adjustment step and is easy to implement in practice. As offshoots of this auction, we also suggest other simple auctions (in which truthful bidding is not an equilibrium by suppliers) which may be suitable where incentives to suppliers are not a big concern. Computer simulations of our auction show that it is scalable for the multi-unit case, and has better information revelation properties than its descending auction counterpart.     

A Note on Existence and Convergence of Best Proximity Points for Pointwise Cyclic Contractions

We introduce a notion of pointwise cyclic contraction T satisfying TA ? B and TB ? A to obtain the existence of a point x ∈ A, such that d(x, Tx) = dist(A, B), known as a best proximity point for such a map. We also prove that for any x ∈ A, the Picard iteration {T²ⁿx} converges to a best proximity point.     

Anaerobic-anoxic-aerobic sequential degradation of synthetic wastewaters

This study was conducted in a continuous three-stage system of anaerobic (R1)-anoxic(R2)-aerobic (R3) reactors with synthetic wastewater containing phenol (1000 mg/L), chemical oxygen demand (COD) (3000 mg/L), CN⁻ (30 mg/L), SCN⁻(400 mg/L), and NH ₄ ⁺ -N (600 mg/L) as principal pollutants and well-acclimated heterogeneous microbial cultures. The final effluent was partially returned to R2 with a recycle ratio of 1. Anaerobic stage served to detoxify the feed by removing up to 80% of cyanide. Complete SCN⁻ removal and denitrification could be achieved in the anoxic stage by utilizing phenol as an internal source of carbon. Nitrification efficiency of 93% was obtained in the aerobic reactor. The results demonstrated that the three-stage system can give the desired final treated effluent quality (0 mg/L of phenol, 0.2 mg/L of CN⁻, 210 mg/L of COD, and 20 mg/L of NH ₄ ⁺ -N) and that the NO ₃ ⁻ -N concentration can be lowered by a higher recycle ratio.     

Fixed-Point Theorems for Generalized Nonexpansive Mappings

Using the concept of asymptotic center we obtain the existence of fixed points having preassigned location for a wider class of asymptotic nonexpansive mappings in a uniformly convex Banach space. This generalization leads us to get a recent result of Alfuraidan and Khamsi for continuous monotone asymptotic nonexpansive mappings as well as the classical fixed-point result of Geobel and Kirk for asymptotic nonexpansive mappings in a uniformly convex Banach space. Also we prove a fixed-point theorem for order preserving continuous maps on a quasiordered closed convex subset of a uniformly convex Banach sapce having monotone norm.     

Modeling and simulation of the CLS cryogenic system

This paper presents results pertaining to the numerical modeling of the cryogenic system at the Canadian Light Source. The cryogenic system consists of a cryostat that houses a Radio Frequency (RF) cavity used for boosting the energy of an electron beam. For consistent operation of the RF cavity, it must be kept immersed in liquid helium at a constant level with the pressure in the gas space maintained to an accuracy of ±1 mbar. An improvement to the cryostat model suggested in [3] using control volumes is described. The model and numerical method developed for the liquid helium supply and gaseous helium return lines are validated using two different cases, viz., the liquid helium flow rate from the liquid helium transfer line and the gaseous helium flow rate from the cryostat for various heater power input settings. The numerical method described here is significantly more accurate, efficient, and flexible than that used in [1] based on an iterative bisection method.