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Base Functions (base + function)
Selected AbstractsDistribution of Aggregate Utility Using Stochastic Elements of Additive Multiattribute Utility ModelsDECISION SCIENCES, Issue 2 2000Herbert Moskowitz ABSTRACT Conventionally, elements of a multiattribute utility model characterizing a decision maker's preferences, such as attribute weights and attribute utilities, are treated as deterministic, which may be unrealistic because assessment of such elements can be imprecise and erroneous, or differ among a group of individuals. Moreover, attempting to make precise assessments can be time consuming and cognitively demanding. We propose to treat such elements as stochastic variables to account for inconsistency and imprecision in such assessments. Under these assumptions, we develop procedures for computing the probability distribution of aggregate utility for an additive multiattribute utility function (MAUF), based on the Edgeworth expansion. When the distributions of aggregate utility for all alternatives in a decision problem are known, stochastic dominance can then be invoked to filter inferior alternatives. We show that, under certain mild conditions, the aggregate utility distribution approaches normality as the number of attributes increases. Thus, only a few terms from the Edgeworth expansion with a standard normal density as the base function will be sufficient for approximating an aggregate utility distribution in practice. Moreover, the more symmetric the attribute utility distributions, the fewer the attributes to achieve normality. The Edgeworth expansion thus can provide a basis for a computationally viable approach for representing an aggregate utility distribution with imprecisely specified attribute weights and utilities assessments (or differing weights and utilities across individuals). Practical guidelines for using the Edgeworth approximation are given. The proposed methodology is illustrated using a vendor selection problem. [source] Simple modifications for stabilization of the finite point methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2005B. Boroomand Abstract A stabilized version of the finite point method (FPM) is presented. A source of instability due to the evaluation of the base function using a least square procedure is discussed. A suitable mapping is proposed and employed to eliminate the ill-conditioning effect due to directional arrangement of the points. A step by step algorithm is given for finding the local rotated axes and the dimensions of the cloud using local average spacing and inertia moments of the points distribution. It is shown that the conventional version of FPM may lead to wrong results when the proposed mapping algorithm is not used. It is shown that another source for instability and non-monotonic convergence rate in collocation methods lies in the treatment of Neumann boundary conditions. Unlike the conventional FPM, in this work the Neumann boundary conditions and the equilibrium equations appear simultaneously in a weight equation similar to that of weighted residual methods. The stabilization procedure may be considered as an interpretation of the finite calculus (FIC) method. The main difference between the two stabilization procedures lies in choosing the characteristic length in FIC and the weight of the boundary residual in the proposed method. The new approach also provides a unique definition for the sign of the stabilization terms. The reasons for using stabilization terms only at the boundaries is discussed and the two methods are compared. Several numerical examples are presented to demonstrate the performance and convergence of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd. [source] Multipole moments and polarizability of molecular systems with D3h symmetry in orbitally degenerate statesINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 6 2006I. Ya. Abstract It was proved, by ab initio studies, that the electronic ground states of the molecules MF3 (MV, Cr, Mn), M3 (MLi, Na, K), and C3H3, with D3h symmetry, have orbital degeneracy. It was shown that in the base functions of these degenerate states, the reduced matrix elements of the in-plane E,-type components of the dipole moment, of the quadrupole moments and of the nontotal symmetric components of the (hyper)polarizability are nonzero. The computed values of the dipole and quadrupole moments of the polarizability and hyperpolarizabilities are comparable to those of the molecular systems of the lower than D3h symmetry. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source] Use of the wavelet method for analyzing electrochemical noise dataMATERIALS AND CORROSION/WERKSTOFFE UND KORROSION, Issue 12 2007O. Duran This research paper shows the application of the mathematical tool "wavelet's" to be used in the interpretation and analysis of electrochemical noise signals. The Fast Fourier Transformation (FFT) is one of the most commonly used techniques for analyzing electrochemical noise. But the application of this methodology to the analysis of stochastic signals does not allow their interpretation in a suitable way. The wavelet's methodology carries out the analysis by means of the construction of finite base functions. In several studies, it has been found out that the function that is closest to the electrochemical oscillations is the one called "coiflets," which presents a similar geometry to what it is observed when an ionic,electronic interchange occurs during an electrochemical interphase. The analysis that appears here was implemented using the toolbox "LabViewÔ." [source] A Novel Tripodal Ligand Containing Three Different N -Heterocyclic Donor Functions and Its Application in Catechol Dioxygenase MimickingCHEMISTRY - A EUROPEAN JOURNAL, Issue 22 2009Marit Wagner Dipl.-Chem. Abstract Prominent donors: A pyridyl, an imidazolyl, and a pyrazolyl donor function are part of the novel tripodal ligand depicted, which thus combines three of the most prominent donors applied in ligands for bioinorganic chemistry within one coordination unit. To exploit its behaviour and potential, first investigations have been carried out in relation to catechol dioxygenase mimicry. We describe a novel chiral ligand, L, in which three different N -donor functions are linked to a methoxymethine unit: a methylpyrazole derivative, a methylimidazole unit, and a pyridyl residue. Complexes with FeCl2, FeBr2, and FeCl3 have been synthesized and fully characterized, including with respect to their molecular structures. While in combination with FeCl3L coordinates in a tripodal fashion, with FeX2 (X=Cl, Br) it binds only through two functions and the pyridyl unit remains dangling. For potential modelling of intradiol and extradiol catechol dioxygenase reactivity, the complexes [LFeCl2], 1, and [LFeCl3], 3, have been treated with 3,5-di- tert -butylcatechol, triethylamine, and O2. Both complexes yielded similar results in such investigations, since the LFeII,catecholate complex reacts with O2 through one-electron oxidation in the first step. Employing 3 in acetonitrile solution, intradiol cleavage occurred, although the undesired quinone was formed as the main product. If reagents were added (NaBPh4, H+) or reaction conditions were chosen (CH2Cl2 instead of CH3CN as the solvent) that made the coordination sphere at the iron centre more accessible for a third substrate donor function, an alternative reaction route, presumably involving O2 binding at the metal, became more important, which led to extradiol cleavage. In the extreme case (CH2Cl2 as the solvent and with the addition of NaBPh4), mainly the extradiol cleavage products were formed; the intradiol products were only observed as side products then and quinone formation became negligible. Protonated base functions in the second coordination sphere increased the efficiency of extradiol cleavage only slightly. The obtained results are in line with current understanding of the function of intradiol/extradiol dioxygenases. [source] | ||||||||||