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EN
The Internet shopping optimization problem arises when a customer aims to purchase a list of goods from a set of web-stores with a minimum total cost. This problem is NP-hard in the strong sense. We are interested in solving the Internet shopping optimization problem with additional delivery costs associated to the web-stores where the goods are bought. It is of interest to extend the model including price discounts of goods. The aim of this paper is to present a set of optimization algorithms to solve the problem. Our purpose is to find a compromise solution between computational time and results close to the optimum value. The performance of the set of algorithms is evaluated through simulations using real world data collected from 32 web-stores. The quality of the results provided by the set of algorithms is compared to the optimal solutions for small-size instances of the problem. The optimization algorithms are also evaluated regarding scalability when the size of the instances increases. The set of results revealed that the algorithms are able to compute good quality solutions close to the optimum in a reasonable time with very good scalability demonstrating their practicability.
EN
Rough set theory has been successfully used in formation system for classification analysis and knowledge discovery. The upper and lower approximations are fundamental concepts of this theory. The new information arrives continuously and redundant information may be produced with the time in real-world application. So, then incremental learning is an efficient technique for knowledge discovery in a dynamic database, which enables acquiring additional knowledge from new data without forgetting prior knowledge, which need to be updated incrementally while the object set get varies over time in the interval-valued ordered information system. In this paper, we analyzed the updating mechanisms for computing approximations with the variation of the object set. Two incremental algorithms respectively for adding and deleting objects with updating the approximations are proposed in interval-valued ordered information system. Furthermore, extensive experiments are carried out on six UCI data sets to verify the performance of these proposed algorithms. And the experiments results indicate the incremental approaches significantly outperform non-incremental approaches with a dramatic reduction in the computational time.
EN
We consider solving the Cauchy problem with an abstract linear evolution equation by means of the Generalized Method of Lie-algebraic discrete approximations. Discretization of the equation is performed by all variables in equation and leads to a factorial rate of convergence if Lagrange interpolation is used for building quasi representation of differential operator. The rank of a finite dimensional operator and approximation properties have been determined. Error estimations and the factorial rate of convergence have been proved.
EN
The lower and upper approximations in rough set theory will change dynamically over time due to the variation of the information system. Incremental methods for updating approximations in rough set theory and its extensions have received much attention recently. Most existing incremental methods have difficulties in dealing with fuzzy decision systems which decision attributes are fuzzy. This paper introduces an incremental algorithm for updating approximations of rough fuzzy sets under the variation of the object set in fuzzy decision systems. In experiments on 6 data sets from UCI, comparisons of the incremental and non-incremental methods for updating approximations are conducted. The experimental results show that the incremental method effectively reduces the computational time.
EN
In this paper we investigate a mathematical model of cancer invasion of tissue, which incorporates haptotaxis, chemotaxis, proliferation and degradation rates for cancer cells and the extracellular matrix, kinetics of urokinase receptor, and urokinase plasminogen activator cycle. We solve the model using spectrally accurate approximations and compare its numerical solutions with laboratory data. The spectral accuracy allows to use low-dimensional matrices and vectors, which speeds up the computations of the numerical solutions and thus to estimate the parameter values for the model equations. Our numerical results demonstrate correlations between numerical data computed from the mathematical model and in vivo tumour growth rates from prostate cell lines.
6
Content available remote Aproksymacja jednostajna odwzorowań kartograficznych
PL
W artykule zaprezentowane zostaną własności wielomianów Czebyszewa oraz ich zastosowanie do aproksymacji odwzorowań kartograficznych. Ponadto przedstawione zostanie porównanie wyników aproksymacji jednostajnej oraz aproksymacji średniokwadratowej odwzorowań kartograficznych.
EN
Usually the least square method is used for approximation of map projection. Determination of polynomial coefficients requires solution of a complicated system of equations. It is possible to avoid such a problem using orthogonal Chebyshev polynomials. This is a completely different method of approximation, where the maximum difference between the value of the function and the value calculated from polynomial is minimized. In the paper, properties of Chebyshev polynomials approximations are presented as wall as their application to map projection approximation and comparison with other methods of map projection.
7
Content available remote On the Kantorovich variant of generalized Bernstein type rational functions
EN
In the present paper we define Kantorovich variant of generalized Bernstein type rational functions. We establish the order of approximation for continuous functions in different normed spaces and also estimate the rate of convergence for functions of bounded variation.
8
Content available remote Random sums stopped by a rare event : a new approximation
EN
The convergence of a geometric sum of positive i.i.d. random variables to an exponential distribution is a well-known result. This convergence provided various and useful approximations in reliability, queueing or risk theory. However, for concrete applications, this exponential approximation is not sharp enough for small values of mission time. So, other approximations have been proposed (Bon and Pamphile (2001), Kalashnikov (1997)). In this paper we propose a new point of view where the exponential approximation appears as a first-order approximation. We consider more general random sums stopped by a rare event, where summands are no more assumed to be independent neither nonnegative. So we give a second-order approximation. As illustration we consider stopping time with negative binomial distribution. This approximation provides a new evaluation tool in reliability analysis of highly reliable systems. The accuracy of this approximation is studied numerically.
EN
The convolution of a function f ∈ Cm(Rk ;R) with a Dirac-sequence gives an approximation of f ∈ C°(Rk ;R) by a sequence of functions fn ∈ C∞(Rk ;R) n ≥ 1 converging uniformly on compact subsets of Rk as well as its derivatives. The paper studies global estimations on Rk , i.e. there exists functions hα ∈ C∞(Rk ; R) and a sequence εα(n) → 0 such that [wzór] for all x ∈ Rk, all | α |≤ m and all n ≥ 1. Special subalgebras of C∞(Rk ;R) are introduced for the study of the dominated behavior of the convolution process. Some extensions for Cm(Ω ;R) are obtained with Ω open subset of Rk.
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