One of great unsolved problems of physical adsorption is the influence of heterogeneity on adsorptive properties of adsorption systems (1). Many theoretical procedures have been proposed in order to describe the heterogeneity effects (2-10). One of the most widely known and often used procedures is the integral representation for the overall adsorption isotherm Otwhere 01 is the local adsorption isotherm, )~(e) is the differential energy distribution, and Q is the range of possible variation of the adsorption energy e. The most interesting problem here is to solve the eq. [1] with respect to the function Z(e), when Ot and 01 are known as analytical expressions.Hobson and Armstron9 (11) were the first who investigated theoretically the Dubinin-Radushkievieh equation (D -R) as the overall adsorption isotherm. The D -R isotherm firstly proposed, and widely verified for porous adsorbents (12), has been found to hold for many non-porous adsorbents (metals, glasses) in the submonolayer range. The source of this success has been theoretically investigated by Hobson (13), Haul and Gottwald (14), Ricca, Medana and Bellardo (15), Sparnaay (8), and Marsh and Rand (16). Their results suggested that the D -R behaviour is due to a particular structure of the adsorbing surfaces.Most advanced theoretical results are due to Misra (9) and Cerofolini (6). Assuming that 01 corresponds to a Langmuir isotherm 0~(e,p)= 1 + --e x p ~-~[2] P they have pointed out though in different ways, that the D -R isotherm is related to special distribution Z (s), which is probably characteristic for the majority of heterogeneous adsorbents. Constant K is connected with the molecular partition function for molecules from the monolayer.These results has been obtained by Misra (9) by solving the Stieltjes transform, which is obtained from eq. [1], using the Langmuir isotherm for 01 (17) and by Heer (18) on the basic of statistical mechanics.Cerofolini ( Above B and Pm are two characteristic constants in the equation D-R. However, the energy s and the minimal adsorption energy s,. are given by the following relations P --.[5] e = -R T l n -~ and sm= RTIn K PmAs it appears from the eqs.[4] and r5] the form of the distribution function X(s) depends on the
In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced.The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.
Abstract.It is shown that any A-firmly, 0 < A < 1 , nonexpansive mapping T: C -> C has a fixed point in C whenever C is a finite union of nonempty, bounded, closed convex subsets of a uniformly convex Banach space.
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