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1

Moments of the weighted Cantor measures

Harding Steven N., Riasanovsky Alexander W. N.

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

256--273

EN

Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying [wzór] where φn : x ↦ (x+n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L2μα[0,1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xmdμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the firstmmoments within uniform error ε in O((loglog(1/ε))·m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2,1/2].

2

On two-parameters generalization of Fibonacci Numbers

Bród D.

Mathematica Applicanda

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2017

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Vol. 45, No. 1

81--92

EN

In this paper we introduce a new two-parameters generalization of Fibonacci numbers – distance s-Fibonacci numbers Fs (k, n). We generalize the known distance Fibonacci numbers by adding an additional integer parameter s. We give combinatorial and graph interpretations of these numbers. Moreover, we present some properties of distance s-Fibonacci numbers, which generalize known properties of classical Fibonacci and Padovan numbers.

PL

W artykule wprowadzono dwuparametrowe uogólnienie klasycznych liczb Fibonacciego - odległościowe s-liczby Fibonacciego. Przedstawiono kombinatoryczne i grafowe interpretacje tych liczb. Pokazane zostały także pewne ich własności, które uogólniają znane własności liczb Fibonacciego i liczb Padovana. Wyznaczona została także funkcja tworząca rozważanego ciągu.

3

Some Parity Statistics in Integer Partitions

Blecher A., Mansour T., Munagi A. O.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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Vol. 63, no. 2

123--140

EN

We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands λ1≥⋯≥λk may be enumerated according to descents λi >λi+1 while tracking the individual parities of λi and λi+1. There are two types of parity levels, E=E and O=O, and four types of parity-descents, E>E, E>O, O>E and O>O, where E and O represent arbitrary even and odd summands. We obtain functional equations and explicit generating functions for the number of partitions of n according to the joint occurrence of the two levels. Then we obtain corresponding results for the joint occurrence of the four types of parity-descents. We also provide enumeration results for the total number of occurrences of each statistic in all partitions of n together with asymptotic estimates for the average number of parity-levels in a random partition.

4

Investigation of G-network with signals at transient behavior

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

75--86

EN

G-queueing network with positive messages and signals at transient behavior is considered. A system of difference-differential equations for the state probabilities of the network is obtained. To find them and the average characteristics of the network a technique was applied based on the use of apparatus of multivariate generating functions. An expression for the generating function was obtained. An example is calculated.

5

Nonstationary analysis of the queueing network with bypass of systems of multi-type and absolute priority messages

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 1

93--105

EN

This paper contains an investigation of an open exponential network with bypass of systems of multi-type messages with absolute priority in transient behavior. Messages with priority received by the node with the probability depending on the state system and the type of incoming messages are for service, forcing messages that are serviced or instant changes in the following system, and can leave the network. Network settings: the intensity of the incoming flow and the intensity of service of different types of messages in single-line systems and the conditional probabilities of transitions of messages between the systems are dependent on the time. We obtained a system difference-differential equations for the state probabilities of the network. To find the state probabilities of the network in the transitional behavior applied a methodology based on the use of the apparatus of multidimensional generating functions.

6

Nonstationary analysis of a queueing network with positive and negative messages

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 2

61--71

EN

This paper contains an investigation of an open queueing network with positive and negative messages that can be used to model the behavior of viruses in information and telecommunication systems and networks. The purpose of research is investigation of such a network at the transient behavior. We consider the case when the intensity of the incoming flow of positive and negative messages and service intensity of messages do not depend on time. It is assumed that all queueing systems of network are one-line. We obtained a system difference-differential equations for the state probabilities of the network. To find the state probabilities of the network in the transitional behavior applied a methodology based on the use of the apparatus of multidimensional generating functions. We obtained an expression for generating function. An example is calculated.

7

Novel digital PWM strategies of multilevel voltage source inverters

Beriber D., Talha A., Berkouk E. M., Francois B., Boucherit M. S.

Archives of Control Sciences

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2009

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Vol. 19, no. 1

43-57

EN

In this paper, two novel algorithms of algebraic PWM strategies for seven-level Neutral Point Clamped (NPC) Voltage Source Inverter (VSI) are presented. These algorithms use a control model of the inverter. In the first part, the authors develop a model of this inverter, without assumption on its control, by using design method associated to Petri nets and switching functions. In the second part, knowledge and control models of this converter are reviewed by using respectively instantaneous and continuous equivalent switching functions. In the last part, two algebraic PWM control strategies of the inverter are proposed. The obtained results are full of promise to use these digital strategies to control seven - level NPC VSI used in high voltage and high power applications as electrical traction.

8

Orthogonal bases for spaces of complex spherical harmonics

Menegatto V. A., Oliveira C. P.

Journal of Applied Analysis

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2005

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Vol. 11, nr 1

113-132

EN

This paper proposes an inductive method to construct bases for spaces of spherical harmonics over the unit sphere Ω 2q of Cq. The bases are shown to have many interesting properties, among them orthogonality with respect to the inner product of L²(Ω 2q). As a bypass, we study the inner product [f,g] = f(D)(g(z))(0) over the space P(Cq) of polynomials in the variables [wzór], in which f(D) is the differential operator with symbol f(z). On the spaces of spherical harmonics, it is shown that the inner product [. , .] reduces to a multiple of the L²(Ω 2q) inner product. Bi-orthogonality in (F(Cq), [. , .] ) is fully investigated.

9

Zastosowanie obliczeń symbolicznych do generowania rozwiązania równania falowego

Ciałkowski M., Jarosławski M.

Zeszyty Naukowe Politechniki Poznańskiej. Maszyny Robocze i Transport

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2003

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Nr 56

115-140

PL

Na podstawie metody rozdzielania zmiennych przedstawiono sposób generowania kolejnych funkcji tożsamościowo spełniających zadane równanie różniczkowe (funkcje Trefftza). Podstawą do wyznaczania tych funkcji jest funkcja generująca zależna od parametru (lub parametrów). Kolejne pochodne funkcji generującej względem parametru wyznaczają kolejne funkcje spełniające zadane równanie różniczkowe. Za pomocą operacji symbolicznych wyznacza się w postaci jawnej poszukiwane funkcje i ich pochodne. Funkcje te mogą być zastosowane jako funkcje bazowe w metodzie elementu skończonego.

EN

The following article presents the description and sphere of activity of a nozzle-flapper system which works as the first part of an amplifier in the hydraulic servovalve. Moreover, on the basis of static characteristics of the amplifier the description and procedures of the parameters choice are described. The methodology was verified by the comparison of results with magnitudes of real parameters of the MOOG 76-225 model servovalve.

10

Generating functions of orthogonal polynomials and Szegö-Jacobi parameters

Asai N., Kubo I., Kuo H.-H.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

273--291

EN

In this paper, we present a more direct way to compute the Szegö-Jacobi parameters from a generating function than that in [5] and [6]. Our study is motivated by the notions of one-mode interacting Fock spaces defined in[1] and Segal-Bargmann transform associated with non-Gaussian probability measures introduced in [2]. Moreover, we examine the relationships between the representations of orthogonal polynomials in terms of differential or difference operators and our generating functions. The connections provide practical criteria to determine when functions of a certain form are orthogonal polynomials.

11

Investigation of probabilistic model of computer network with central servicing system in transient behavior

Malatycki M., Pankow A.

Informatyka Teoretyczna i Stosowana

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2003

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R. 3, nr 4

7-18

EN

This paper provides the method of generating functions using for calculating the time-dependent state probabilities for open networks in transient time.

12

Integrals of legendre polynomials and solution of some partial differential equations

Belinsky R.

Journal of Applied Analysis

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2000

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Vol. 6, nr 2

259-282

EN

We show a connection between the polynomials whose inflection points coincide with their interior roots (let us write shorter PIPCIR), Legendre polynomials, and Jacobi polynomials, and study some properties of PIPCIRs (Part I). In addition, we give new formulas for some classical orthogonal polynomials. Then we use PIPCIRs to solve some partial differential equations (Part II).