Triple solutions for a nonlocal functional boundary value problem by leggett–williams theorem

The existence of triple solutions for a second-order nonlocal functional boundary value problem is proved by using a fixed-point theorem on cones in Banach spaces due to Leggett and Williams. The obtained results are new even in the ordinary case for three-point boundary value problems discussed quite recently in [Xiaoming He and Weigao Ge (2002). Triple solutions for second-order three-point boundary value problems. J. Math. Anal. Appl., 268, 256–265].     

Pricing vulnerable European options with stochastic default barriers

CF Lo and KC Ku Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Email: cho-hoi_hui{at}hkma.gov.hk Received on 31 July 2006. Accepted on 15 March 2007. This paper develops a valuation model of European options incorporatinga stochastic default barrier, which extends a constant defaultbarrier proposed in the Hull–White model. The defaultbarrier is considered as an option writer's liability. Closed-formsolutions of vulnerable European option values based on themodel are derived to study the impact of the stochastic defaultbarriers on option values. The numerical results show that negativecorrelation between the firm values and the stochastic defaultbarriers of option writers gives material reductions in optionvalues where the options are written by firms with leverageratios corresponding to BBB or BB ratings.     

Approximation algorithms for a vehicle routing problem

In this paper we investigate a vehicle routing problem motivated by a real-world application in cooperation with the German Automobile Association (ADAC). The general task is to assign service requests to service units and to plan tours for the units such as to minimize the overall cost. The characteristics of this large-scale problem due to the data volume involve strict real-time requirements. We show that the problem of finding a feasible dispatch for service units starting at their current position and serving at most k requests is NP-complete for each fixed k ≥ 2. We also present a polynomial time (2k − 1)-approximation algorithm, where again k denotes the maximal number of requests served by a single service unit. For the boundary case when k equals the total number |E| of requests (and thus there are no limitations on the tour length), we provide a -approximation. Finally, we extend our approximation results to include linear and quadratic lateness costs.     

From hypervalent xenon difluoride and aryliodine(III) difluorides to onium salts: Scope and limitation of acidic fluoroorganic reagents in the synthesis of fluoroorgano xenon(II) and iodine(III) onium salts

Fluorinated organodifluoroboranes R_fBF₂ are in general suitable reagents to transform XeF₂ and RIF₂ into the corresponding onium tetrafluoroborate salts [R_fXe][BF₄] and [R(R_f)I][BF₄], respectively. (4-C₅F₄N)BF₂ and trans-CF₃CFCFBF₂ which represent boranes of high acidity form no Xe-C onium salts in reactions with XeF₂ but give the desired iodonium salts with RIF₂ (R = C₆F₅, o-, m-, p-C₆FH₄). The reaction of (4-C₅F₄N)BF₂ with XeF₂ ends with a XeF₂-borane adduct. C₆F₅Xe(4-C₅F₄N), the first Xe-(4-C₅F₄N) compound, was obtained when C₆F₅XeF was reacted with Cd(4-C₅F₄N)₂. We describe the synthesis of (4-C₅F₄N)IF₂ and reactions of (4-C₅F₄N)IF₂ and C₆F₅IF₂ with (4-C₅F₄N)BF₂. Analogous to [(4-C₅F₄N)₂I][BF₄] and [C₆F₅(4-C₅F₄N)I][BF₄] aryl(perfluoroalkenyl)iodonium salts [R(R′)I][BF₄] were obtained from RIF₂ (R = C₆F₅, o-, m-, p-C₆FH₄) and R′BF₂ (R′ = trans-CF₃CFCF, CF₂CF). The gas phase fluoride affinities pF⁻ of selected fluoroorganodifluoroboranes R_fBF₂ and their hydrocarbon analogs are calculated (B3LYP/6-31+G^*) and discussed with respect to their potential to introduce R_f-groups into hypervalent EF₂ bonds. Four aspects which influence the transformation of hypervalent EF₂ bonds (E = Xe, R′I) under the action of Lewis acidic reagents RAF_n−1 (A = B, P; n = 3, 5) into the corresponding [RE][AF_n+1] salts are presented and the important role of the acidity is emphasized. Fluoride affinities may help to plan the introduction of organo groups into EF₂ moieties and to expand the types of acidic reagents. Thus C₆H₅PF₄ with a pF⁻ value comparable to that of R_fBF₂ compounds is able to introduce the C₆H₅ group into RIF₂ (R = C₆F₅, p-C₆FH₄).     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.     

ChemInform Abstract: Glass Formation in the System BaS-Ga2S3-GeS2 and the Structure of Ba2. 7Ga5.4Ge3.6S18.

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