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1

Application of polynomial tensor interpolation for technical tests

Ciekot Anita, Biernat Grzegorz, Frączek Tadeusz

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 1

17--24

EN

The main aim of this paper is a new formula of tensor interpolation by the polynomial of two variables. The formulas for interpolating polynomial coefficients are obtained using the Kronecker tensor product of matrices. The mathematical model for the diffusion process is presented. This paper is focused on determining the optimal parameters for this process by polynomial tensor interpolation of the obtained research results.

2

Optimal Parisian-Type Dividend Payments Penalized by the Number of Claims for the Classical and Perturbed Classical Risk Process

Czarna Irmina, Li Yanhong, Palmowski Zbigniew, Zhao Chunming

Probability and Mathematical Statistics

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2020

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Vol. 40, Fasc. 1

57--81

EN

We consider the classical risk process (the case σ = 0) and the classical risk process perturbed by a Brownian motion (the case σ > 0). We analyze the expected NPV describing the mean of the cumulative discounted dividend payments paid up to the Parisian or classical ruin time and further penalized by the number of claims that appeared up to that time. We identify this function for a constant barrier strategy and we find sufficient conditions for this strategy to be optimal. We also analyze a numerical example of exponential claim sizes.

3

Echoes and glimpses of a distant drum

Turner John Wm.

Opuscula Mathematica

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2020

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Vol. 40, no. 6

737-–750

EN

To what extent does the spectrum of the Laplacian operator on a domain D with prescribed boundary conditions determine its shape? This paper first retraces the history of this problem, then Kac’s approach in terms of a diffusion process with absorbing boundary conditions. It is shown how the restriction to a polygonal boundary for D in this method, which required taking the limit of an infinite number of sides to obtain a smooth one, can be avoided by using the Duhamel method.

4

Mathematical modeling of random concentracion field and its second moments in semispace with erlangian distributions of layerd inclusions

Chernukha O., Bilushchak Y.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2016

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Vol. 20, No 3

295--334

EN

The processes of admixture diffusion in a two-phase stratified semispace with random disposition of syblayers are studied by the approach where internal random nonhomogeneities are considered as inner sources and the solution is found in the form of a Neumann series. The diffusion equations are formulated for one-connected regions of each phase and non-ideal contact conditions for the concentration on interphases are imposed. By the theory of generalized functions the contact problem is reduced to the equation of mass transfer in the whole body, which operator includes explicitly jump discontinuities of the concentration function and its derivatives. The obtained initial-boundary value problem of mass transfer is reduced to the equivalent integro-differentual equation. The solution is constructed in the form of a Neumann series and averaged over the ensemble of phase configurations with Erlangian and exponential distributions of inclusions. Dispersion and the two-point correlation function of the concentration field for diffusion are determined taking into account the probable distribution of inclusions, pair interaction of sublayers and the function of phase correlation. The dependence of the behavior of the averaged admixture concentration, field dispersion and the correlation function in the semispace with Erlangian and exponential distributions of inclusions on different medium characteristics is investigated and established.

5

One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary

Kopytko B., Shevchuk R.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

71-82

EN

By the method of the classical potential theory, we construct the two-parameter Feller semigroup of operators associated with such a diffusion phenomenon on a half-line with a moving boundary where either a reflection or jump phenomenon occurs at a boundary point.

6

Simulation of Diffusion Flows in Two-Phase Multilayered Stochastically Nonhomogeneous Bodies with Non-Uniform Distribution of Inclusions

Chaplya Y., Chernukha O., Davydok A.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2015

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Vol. 19, No 3

321--351

EN

Admixture diffusion flows are investigated in two-phase randomly nonhomogeneous multilayered strips with non-uniform distributions of inclusions. Cases where the most probable disposition of layered inclusions is located near the body boundary on which the mass source acts in the neighborhood of another boundary and in the middle of the body are considered. The initial-boundary value problem is formulated for the function of random mass flow under conditions of a constant flow on the upper surface and zero concentration of the admixture on the lower surface. Calculation formulae are obtained for the diffusion flow averaged over the ensemble of phase configurations in the particular cases of beta-distribution at zero and nonzero initial concentrations. The dependences of the averaged admixture flows on medium characteristics are established. It is shown that if the admixture diffusion coefficient in inclusions is greater than in the matrix, consolidation of inclusions in the middle of the body leads to an increasing diffusion flow. Simulation of the averaged diffusion flows of the admixture in the multilayered strip is performed for different model variants of a probable disposition of phases in the body and their comparative analysis is carried out.

7

Mathematical Modeling of Random Diffusion Flows in Two-Phase Multilayered Stochastically Nonhomogeneous Bodies

Chaplya Y., Chernukha O., Davydok A.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2015

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Vol. 19, No 3

297--320

EN

An approach for studying stochastical diffusion flows of admixture particles in bodies of multiphase randomly nonhomogeneous structures is proposed, according to which initialboundary value problems of diffusion are formulated for flow functions and methods of solution construction are adapted for the formulated problems. By this approach the admixture diffusion flow is investigated in a two-phase multilayered strip for the uniform distribution of phases under conditions of constant flow on the upper surface and zero concentration of admixture on the lower surface. An integro-differential equation equivalent to the original initial-boundary value problem is constructed. Its solution is found in terms of the Neumann series. Calculation formulae are obtained for the diffusion flow averaged over the ensemble of phase configurations under both zero and constant nonzero initial concentrations. Software is developed, a dependence of averaged diffusion flows on the medium characteristics is studied and general regularities of this process are established.

8

One-dimensional diffusions in bounded domains with a possible jump-like exit from a sticky boundary

Kopytko B., Shevchuk R.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 3

101--114

EN

Using analytical methods we obtain the integral representation of a two-parameter Feller semigroup on a closed interval [r1, r2] corresponding to such a diffusion phenomenon that sticking, partial reflection, absorption and jump phenomena occur at the endpoints r1, r2 and at some interior point r ∈ (r1,r2).

9

The Structural Stability of the Fe36 Co36 Si19 B5 Nb4 Bulk Amorphous Alloy

Nabiałek M., Pietrusiewicz P., Szota M., Dobrzańska-Danikiewicz A., Lesz S., Dośpiał M., Błoch K., Oźga K.

Archives of Metallurgy and Materials

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2014

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Vol. 59, iss. 1

259--262

EN

In this paper, the results of the investigation into the fractured surface microstructure of the amorphous samples of Fe36 Co36 Si19 B5 Nb4 in the shape of rods of diameters: 1 mm, 2 mm and 3 mm in the as-cast state are presented. The samples were prepared by injection of molten alloy into cooled copper dies. The process of diffusion in the investigated material has a different speed depending on the temperature gradient within the volume of the rod. The atomic diffusion leads to the creation of different zones within the rod fracture: the zone in contact with the copper die, the intermediate fracture zone, and the zone in the vicinity of the rod core; the three zones have been found to exhibit different amorphous structures.

PL

W pracy przedstawiono wyniki badań mikrostruktury na powierzchni przełomów próbek Fe36 Co36 Si19 B5 Nb4 amorficznych w postaci prętów o średnicy 1 mm. 2 mm i 3 mm. w stanie po zestaleniu. Pręty wytworzono metodą wtłaczania ciekłego stopu do miedzianej, chłodzonej cieczą formy. Proces dyfuzji atomów w badanym materiale charakteryzuje się inną dynamiką w zależności od gradientu temperatury w objętości pręta. Wyróżnia się trzy wyraźnie widoczne strefy: od kontaktu z miedzianą formą, strefę przejściową oraz strefę opisującą rdzeń pręta. Każdą z wyróżnionych stref charakteryzują różne konfiguracje atomów w zakresie stanu amorficznego. Na podstawie badań stwierdzono, że w masywnych materiałach amorficznych występuje fluktuacja składu oraz gęstości w zależności od szybkości chłodzenia.

10

The Temperature of the Beginning of Alloy Layer Forming Process on the Steel Cast

Szajnar J., Walasek A., Baron C.

Archives of Foundry Engineering

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2013

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Vol. 13, iss. 1 spec.

185--188

EN

In this work the results of experiment connected with the problem of formation surface layers on the steel castings. There was an attempt to determine two factors: temperature (premould be higher up 1300°C) and time (heating composite premould). The purpose of the argument was prove of creation surface composit layer in the consequence of heat composite premould (to above temperature of 1300°C). The purpose of conducted research was tests control parameters (have an influence on diffusion process).

11

Feller semigroup for diffusion process piesewise-constant generalized diffusion matrix and generalized drift vector

Kopytko B., Novosyadlo A.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 4

75-84

EN

We obtain an integral representation of the classical solution of the conjugation problem for the second order parabolic equation with derivatives with respect to tangent variables at the conjugation conditions. Using this solution, we construct the Feller semigroup to which there corresponds a diffusion process with a piecewise-constant generalized diffusion matrix and a generalized drift vector.

12

Application of the molecular dynamics method for modelling of mass transfer on the border of Ni-Al bimetal

Starostenkov M., Demina I., Popova G., Denisova N., Smolarz A.

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2012

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nr 1

36-38

EN

One of the basic issues of examination of process of interaction of two-component alloys is the study of the mass transfer process in the presence of dot flaws: internodal atoms and vacancies. The study of this process in real experiments is impossible; therefore it is reasonable to apply computer modelling. The authors present a model of Ni-Al bimetal diffusion process produced by the method of molecular dynamics.

PL

Jednym z podstawowych zagadnień analizy procesu interakcji stopów dwuskładnikowych jest badanie procesu wymiany masy w obecności defektów punktowych: atomów międzywęzłowych i wakansów. Badania tego procesu w rzeczywistych eksperymentach jest niemożliwe, dlatego uzasadnione jest zastosowanie modelowania komputerowego. Autorzy przedstawiają model procesu dyfuzji w bimetalu Ni-Al stworzony metodą dynamiki molekularnej.

13

Modelling of mass transport in watercourses at unsteady states

Bielski A.

Environment Protection Engineering

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2012

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Vol. 38, nr 3

159--173

EN

In terms of quality particularly difficult to describe are processes of mass exchange between different phases (e.g., atmospheric air-water, water-river sediment, water-algae, etc.). Whitman's model is most often used to describe the mass transport processes through the phase boundary. Theoretical analysis of the mass transfer process through the phase boundary showed that in unsteady states, the calculation results obtained from Whitman's model differ from the results obtained using the accurate diffusion model. These differences are due to the fact that concentration profiles in the direction of diffusion process change in time. Assumptions for Whitman's model do not include changes in the concentration distribution over time. Therefore, the correction factor was introduced to Whitman's model. The correction factor is defined as a parameter that multiplies a concentration derivative over time in the mass transport model. The correction factor can be used to estimate the effective diffusion coefficient of the substance that permeates from the aqueous phase to the sediment. The correction factor improves the degree of fit of the mass transport model to the measurement data. It can be used to estimate the effective turbulent diffusion coefficient from water phase to the sediment phase.

14

Probabilistic solution to a two-dimensional LQG homing problem

Makasu C., Lefebvre M.

Control and Cybernetics

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2012

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Vol. 41, no. 1

5-12

EN

Let xt be an arbitrary one-dimensional diffusion process and yt be a one-dimensional controlled diffusion process starting from y0 = y ∈ (a, b). The process is controlled until yt crosses either y = a or y = b for the first time. Our aim is to find the control ut that minimizes an expected cost functional with both quadratic control and boundary crossing costs. An explicit form for the optimal control is obtained under certain conditions.

15

Monte Carlo methods of estimating transition density and likelihood function of diffusion processes

Borkowski P., Mielniczuk J.

Prace Instytutu Podstaw Informatyki Polskiej Akademii Nauk

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2009

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Nr 1013

1-23

EN

The properties of various Monte Carlo schemes for estimating transition density of discretely observed diffusion processes are discussed. The considered methods include various importance samplers with weighing distribution taking into account observed trajectory of the process. They are then used to evaluate simulated likelihood function and maximum likelihood estimators.

PL

W pracy rozważane są różne warianty metody Monte Carlo dla estymacji gęstości prawdopodobieństwa przejścia na podstawie procesu dyfuzji obserwowanego w dyskretnych momentach czasowych. W szczególności bada się ważoną metodę Monte Carlo, w której stosowane wagi zależą od obserwowanej trajektorii procesu. Estymatory gęstości przejścia są następnie wykorzystane do estymacji funkcji wiarogodności i estymatorów największej wiarogodności.

16

Some remarks on the maximum of a one-dimensional diffusion process

Abundo M.

Probability and Mathematical Statistics

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2008

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Vol. 28, Fasc. 1

107--120

EN

For a certain class of one-dimensional diffusions X(t); we study the distribution of maxtЄ[0;T] X(t) and the distribution of the first instant at whichX(t) attains the maximum by reducingX(t) to Brownian motion. Moreover, for T fixed or random, we study the asymptotics of threshold crossing probability, i.e. the rate of decay of P(maxsЄ[0;T ] X(s) > z ) as z goes to infinity. Some examples are also reported.

17

Materiały na osnowie faz międzymetalicznych z układu Fe-Al z udziałem Al_2O_3

Karczewski K., Jóźwiak S., Bojar Z., Durejko T.

Archiwum Odlewnictwa

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2003

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R. 3, nr 7

301-312

PL

Tworzywa konstrukcyjne na bazie faz międzymetalicznych ze względu na swoje właściwości są perspektywicznymi materiałami konstrukcyjnymi mogącymi mieć specyficzne zastosowania. Ich cechy użytkowe są silnie zależne od składu chemicznego, a tym samym od budowy fazowej otrzymanego stopu. W pracy przedstawiono możliwości otrzymywania półfabrykatów niejednorodnych materiałów na bazie faz międzymetalicznych z układu Fe-Al z wydzieleniami tlenków Al_O_3. Analizowano wpływ sposobu wytwarzania na budowę fazową otrzymanych tworzyw. Określono zmiany struktury i właściwości poprzez mikroanalizę składu chemicznego oraz pomiary mikrotwardości.

EN

Fe-Al based intermetallics, thanks for their interesting functional properties, are counted into group of new structural materials for use in special conditions (for example in balIistic shield). Their properties are strongly dependent on chemical composition and phase structure building. A practice of preparation of Fe-Al based intermetalIic heterogeneous materials was presented in this paper. An influence of manufacturing method and parameters of technology on the structure phase of obtained materials was analysed. On the basis of results of microstructure and chemical analysis as well microhardness measurement the chan ges of functional properties of Fe-Al based heterogeneous intermetallics were described and discussed.

18

Tail probabilities for a risk process with subexponential jumps in a regenerative and diffusion environment

Palmowski Z.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 2

381-405

EN

In this paper we find a nonexponential Lundberg approximation of the ruin probability in a Cox model, in which a governing process has a regenerative structure and claims are light-tailed or have an intermediate regularly varying distribution. Examples include an intensity process being reflected Brownian motion, square functions of the Ornstein-Uhlenbeck process and splitting reflected Brownian bridges. In particular, we consider a non-Markovian intensity proces.

19

Structure and properties of FeAl sinters resistant to abrassive wear

Durejko T., Bojar Z., Jóźwiak S.

International Journal of Applied Mechanics and Engineering

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2002

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Vol. 7, Spec. is

347-350

EN

A method of obtain of sintered intermetallic alloys by pressing under cyclic loading at elevated temperature has been presented in this paper. Investigated samples were prepared as results of four different value of cyclic loading frequency of manufacturing process. Results of structure and phase analysis, especially in range of a content and morphology of aluminium oxides have been discussed. Basic mechanical properties of sinters estimated by microhardness measuring. On the basic of obtained results it has been shown that FeAl based intermetallic sinters may be treated as chemically and structurally heterogeneous materials with minimal content of pores.

20

Model aproksymacyjny procesu dyfuzji z reakcją chemiczną - zastosowanie do modelowania reaktora heterogenicznego

Szulkiewicz M.

Inżynieria Chemiczna i Procesowa

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2001

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T. 22, z. 3E

1363--1368

PL

W pracy przedstawiono sposób wykorzystania aproksymacyjnego modelu ziarna katalizatora dla izotermicznego procesu opisanego nieliniowym równaniem kinetycznym. Przeprowadzone obliczenia (przy wykorzystaniu różnego typu równań kinetycznych, w szerokim zakresie zmian wartości parametrów modelu) potwierdziły przydatność modelu aproksymacyjnego. Opracowano także kryterium pozwalające określić możliwość wykorzystania modelu aproksymacyjnego na podstawie warunków prowadzenia procesu i równania kinetycznego.

EN

An application of approximate model of catalyst pellet for modeling of fixed-bed reactor is presented. Die calculations which have been carried out confirm high usefulness of the proposed model. For practical purposes, a criterion of evaluation whether LDF formula can be use with sufficiently small error was presented.