Cost allocation in a bank ATM network

We consider a situation in which a group of banks consider connecting their Automated Teller Machines (ATMs) in a network, so that the banks customers may use ATMs of any bank in the network. The problem studied is that of allocating the total transaction costs arising in the network, among the participating banks. The situation is modeled as a cooperative game with transferable utility. We propose two allocations, and discuss their relation to the core and other well-known solution concepts, as well as to population monotonicity.Endre Bjørndal has enjoyed the hospitality of Tilburg University, and has also received financial support from Telenor AS and the Norwegian School of Economics and Business Administration.     

Long Excursions of a Random Walk

In Csáki et al. ⁽¹⁾ and Révész and Willekens⁽⁹⁾ it was proved that the length of the longest excursion among the first n excursions of a plane random walk is nearly equal to the total sum of the lenghts of these excursions. In this paper several results are proved in the same spirit, for plane random walks and for random walks in higher dimensions.     

In-situ mechanical characterization of wurtzite InAs nanowires

High aspect ratio vertical InAs nanowires were mechanically characterized in a scanning electron microscope equipped with two micromanipulators. One, equipped with a calibrated atomic force microscope probe, was used for in-situ static bending of single nanowires along the 〈11–20〉 crystallographic direction. The other one was equipped with a tungsten tip for dynamic resonance excitation of the same nanowires. This setup enabled a direct comparison between the two techniques. The crystal structure was analyzed using transmission electron microscopy, and for InAs nanowires with a hexagonal wutzite crystal structure, the bending modulus value was found to BM=43.5 GPa. This value is significantly lower than previously reported for both cubic zinc blende InAs bulk crystals and InAs nanowires. Besides, due to their high resonance quality factor (Q>1200), the wurtzite InAs nanowires are shown to be a promising candidate for sub-femtogram mass detectors.     

Studies in the Field of Pyridazine Compounds. X1. A Method for the Synthesis of 6-(Substituted Amino)-3-Pyridazinyl-Hydrazines

Recently, the 6-(substituted amino)-3-pyridazinyl-hydrazines^2–4 and their hydrazones¹ became important as peripheral antihypertensive agents. These compounds have been synthesized from 3,6-dichloropyridazine (1) by introducing to the molecule firstly the substituted amine and then the hydrazine moiety^2–4. This process has some disadvantages. It is known that one chlorine atom of 1 can be easily replaced by nucleophilic substitution. The subsequent replacement of the other chlorine atom in 1 is, however, rather difficult⁵. For example, 2 silosequent substitution with hydrazine should be uarried out with agreat excess of hydrazine at elevated temperatures and the product formed is liable to decompose in this basic medium. As a consequence, overail yield of this process is low ana in general, the final product nas to be purifiea tnroilgh its benzylidene derivative. Tnerefore, according to our experiments, this process does not seem to be useful for large-scale operations.     

Enhanced RFB method

The residual-free bubble method (RFB) is a parameter-free stable finite element method that has been successfully applied to a wide range of boundary-value problems exhibiting multiple-scale behaviour. If some local features of the solution are known a priori, the approximation properties of the RFB finite element space can be improved by enriching it on selected edges of the partition by edge-bubbles that are supported on pairs of neighbouring elements. Motivated by this idea, we define and analyse an enhanced residual-free bubble method for the solution of convection-dominated convection-diffusion problems in 2-D. Our a priori analysis highlights the limitations of the RFB method and the improved global approximation properties of the new method. The theoretical results are supported by detailed numerical experiments.     

IR, 1H NMR and 13C NMR spectra of isomeric 1-aryl-4-methyl-3- and-5-pyrazolecarboxylic esters

The IR, ¹H and ¹³C NMR spectra of six esters of 1-aryl-4-methyl-3- and 5-pyrazolecarboxylic acids are reported. On the basis of their spectra the 3- or 5-position of the substituent in such structural isomeric pairs can be established without recourse to the spectrum of the other member of the pair. The spectral data also give information on the conformation, which depends on the relative position and size of the hetero ring substituents.     

Structural Approach to Subset Sum Problems

We discuss results obtained jointly with Van Vu on the length of arithmetic progressions in \(\ell \)-fold sumsets of the form

$$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\} \end{aligned}$$

and

$$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\text { all distinct}\}, \end{aligned}$$

where \(\mathcal {A}\) is a set of integers. Applications are also discussed.

     

Logical analysis of data: classification with justification

Learning from examples is a frequently arising challenge, with a large number of algorithms proposed in the classification, data mining and machine learning literature. The evaluation of the quality of such algorithms is frequently carried out ex post, on an experimental basis: their performance is measured either by cross validation on benchmark data sets, or by clinical trials. Few of these approaches evaluate the learning process ex ante, on its own merits. In this paper, we discuss a property of rule-based classifiers which we call “justifiability”, and which focuses on the type of information extracted from the given training set in order to classify new observations. We investigate some interesting mathematical properties of justifiable classifiers. In particular, we establish the existence of justifiable classifiers, and we show that several well-known learning approaches, such as decision trees or nearest neighbor based methods, automatically provide justifiable classifiers. We also identify maximal subsets of observations which must be classified in the same way by every justifiable classifiers. Finally, we illustrate by a numerical example that using classifiers based on “most justifiable” rules does not seem to lead to overfitting, even though it involves an element of optimization.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et?al. (J. Non-Newtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of high-dimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson’s equation on a rectangular domain in ?², subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30:621–651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein–Uhlenbeck operator of the kind that appears in Fokker–Planck equations that arise in bead–spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D=D ₁×?×D _N contained in ? ^Nd , where each set D _i , i=1,…,N, is a bounded open ball in ? ^d , d=2,3.