On pitfalls in computing the geodetic number of a graph

Given a simple connected graph G = (V, E) the geodetic closure of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes . The geodetic number, denoted by g(G), is the smallest cardinality of a node set S ^* such that I[S ^*] = V. In “The geodetic number of a graph”, [Harary et al. in Math. Comput. Model. 17:89–95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0–1 integer programming model to find the geodetic number. Computational results are given.     

Well-posedness results for differential and integral equations containing Henstock–Kurzweil integrable vector-valued functions

In this paper, we prove existence, uniqueness and comparison results for solutions of differential and integral equations in Banach spaces containing Henstock–Kurzweil integrable functions from a compact real interval to an ordered Banach space.     

Radiation sintering of magnesia: Pore structure studies

Co⁶⁰ γ-ray irradiation of a nonequilibrium surface of magnesium oxide produced tremendous effects on both the extent of the surface area, and on the pore structure. In all cases investigated incompletely decomposed crystals showed appreciable lowering of surface area which indicated activated sintering of the material. The pores present are mainly wide, but with a very narrow distribution of their sizes. Their most probable hydraulic radius is about 10 Å. Upon irradiation sintering is accompanied by widening of pores, and a second group of pores showed its appearance in the pore size distribution curves. These effects seem to be associated with the presence of traces of the volatile component of the parent material, and in particular with the presence of traces of water vapor. Irradiation of the completely decomposed material produced no changes in surface area or pore structure.

The role played by water vapor in accelerating the sintering of magnesia is discussed and possible explanations are given for the effects of irradiation. Dual effects of water vapor might lead to the sintering acceleration of magnesia; chemisorption of OH groups which create cation vacancy on the surface leading to activated surface diffusion, and oxygen bridging between two adjacent/or opposite OH groups particularly in narrow pores, leading to their blocking.     

Influence of random electric fields on luminescence yield and kinetics of insulators

In the present work we investigate theoretically the influence of random electric fields on electron-hole recombination in wide bandgap crystals. Effective Onsager radius and, therefore, electron-hole recombination rate are significantly modified by external electric fields. Electric field distribution functions for point defects and charged dislocations are evaluated analytically. Electron-hole recombination rate decreases with concentration of point defects and dislocations. In simple case of random fields created by charge carriers in highly excited regions the recombination rate is proportional to n 2/3 rather than n , where n is the concentration of excitations. Therefore modification of luminescence kinetics is most pronounced at initial stages of relaxation of highly excited regions.     

Target heating during ion implantation and related problems

We have undertaken a general analysis of wafer heating during implantation and first, we present general considerations of this problem with respect to amorphization doses extracted from Morehead and Crowder theory. Then, radiative properties of silicon wafers are measured: a law of variation of the emissivity with temperature is given; a comparison is made with values found in the literature. Dependence with experimental conditions (heat reflector, conductive losses) is also studied. Three methods for temperature measurement are used: temperature coefficient of a small resistor vacuum deposited on a face, thermocouple measurement and infrared detection. We compare the three kinds of results which are in good agreement and we make some comments about the temperature measurements using infrared detection.     

浊点萃取-分光光度法测定彩笔中的灿烂绿

建立了浊点萃取-分光光度法测定灿烂绿的方法.以非离子型表面活性剂Triton X-114为萃取剂,富集、分离灿烂绿,采用分光光度法进行检测.研究了缓冲溶液用量、表面活性剂用量、平衡温度和平衡时间对浊点萃取的影响,得到最佳实验条件：5％的Triton X-114溶液2.5mL、pH3.8的盐酸-柠檬酸钠缓冲溶液2.0mL、平衡温度和时间分别为50℃和15min.在优化的实验条件下,灿烂绿被萃取到Triton X-114相与水相分离.该方法用于彩笔中灿烂绿的测定,结果满意.     

Best proximity point theorems

Let us assume that A and B are non-empty subsets of a metric space. In view of the fact that a non-self mapping T:A?B does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element x that is as close to Tx as possible. In other words, when the fixed point equation Tx=x has no solution, then it is attempted to determine an approximate solution x such that the error d(x,Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, known as best proximity points, of the fixed point equation Tx=x when there is no solution. Because d(x,Tx) is at least d(A,B), a best proximity point theorem ascertains an absolute minimum of the error d(x,Tx) by stipulating an approximate solution x of the fixed point equation Tx=x to satisfy the condition that d(x,Tx)=d(A,B). This article establishes best proximity point theorems for proximal contractions, thereby extending Banach’s contraction principle to the case of non-self mappings.     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

Fixed points, selections and common fixed points for nonexpansive-type mappings

We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings.