Evaluation of the Accident Rate of a Plant Equipped with an Aging Single Protective Channel by the Method of Supplementary Variables

This work calculates the reliability of protective systems of industrial facilities, such as nuclear, to analyze the case of equipment subject to aging, important in the extension of the qualified life of the facilities. By means of the method of supplementary variables, a system of partial and ordinary integral-differential equations was developed for the probabilities of a protective system of an aging channel. The system of equations was solved by finite differences. The method was validated by comparison with channel results with exponential failure times. The method of supplementary variables exhibits reasonable results for values of reliability attributes typical of industrial facilities.

Some New Newton’s Type Integral Inequalities for Co-Ordinated Convex Functions in Quantum Calculus

Some recent results have been found treating the famous Simpson’s rule in connection with the convexity property of functions and those called generalized convex. The purpose of this article is to address Newton-type integral inequalities by associating with them certain criteria of quantum calculus and the convexity of the functions of various variables. In this article, by using the concept of recently defined q1q2 -derivatives and integrals, some of Newton’s type inequalities for co-ordinated convex functions are revealed. We also employ the limits of q1,q2→1− in new results, and attain some new inequalities of Newton’s type for co-ordinated convex functions through ordinary integral. Finally, we provide a thorough application of the newly obtained key outcomes, these new consequences can be useful in the integral approximation study for symmetrical functions, or with some kind of symmetry.

From fractional order equations to integer order equations

The main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

ESTIMATION OF POWER-LAW EXPONENT OF DEGREE DISTRIBUTION USING MEAN VERTEX DEGREE

The power-law exponent γ describing the form of the degree distribution of networks is related to the mean degree of the networks. This relation provides a useful method to estimate this exponent because mean degrees can easily be obtained by the number of vertices and edges in the networks. We improved the relation obtained only by ordinary integral approximation, and examined its availability for number of vertices, minimum degree considered, and range of γ, taking also into consideration the number of vertices with minimum degree. As a result, we were able to estimate slight deviations of γ from 3, which are usually observed in numerical simulations of growing networks with linear preferential attachment. Furthermore, using this method, we were also able to predict to what extent γ changed by joining pre-existing vertices in growing networks or by imposing restrictions to prevent the gaining of new edges for certain vertices. For cases where γ < 2, we estimated the power-law exponent of degree distributions of networks formed by traces of random walkers from the increased rate of vertices with created edges.

On a residue of complex functions in the three-dimensional Euclidean complex vector space

In the introductory part of this paper, a notion of absolute integral sums of a complex function, which is more general with respect to that of an integral and integral sums of ordinary integral calculus, is defined. Throughout the main part of the paper, an attempt has been made to generalize, on the basis of redefining the notion of a complex function residue, some of the fundamental results of Cauchy's calculus of residues of analytic functions. The foundation stone of the whole theory is the total value of an improper integral of complex functions.

The Exponent of the Homotopy Groups of Moore Spectra and the Stable Hurewicz Homomorphism

This paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.