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EN
The paper presents a method for investigating the electromagnetic transients of partially under-grounded high voltage overhead transmission lines in the presence of corona. It is based on a time domain model in which the corona is simulated by distributed voltage-dependent shunt current sources. The line segments and the inserted cable section are represented by a single equivalent transmission element having location-dependent circuit parameters per unit length. The numerical solution of the resulting set of differential equations yields the distributions of the voltage, the longitudinal current and the shunt corona current (per unit length) as functions of location and time. The presented two-, three-dimensional and contour plots proved to be helpful in discussing these distributions and in identifying any eventual current and/or voltage concentrations. The developed computer code in Mathematica can handle any time waveform of the source initiating the transients, any line termination as well as any lengths of the overhead line and cable section.
EN
One of most common natural catastrophes in Poland are undoubtedly floods. Climatic change contributes to more and more often and violent occurrences of the maximum flow in rivers, which increases flood damage. Inadequate land management and the unjustified belief in the effectiveness of technical flood control measures can also contribute to flood damage. The development of water management (including flood protection) should be carried out in a sustainable way by integrating social, environmental, and economic objectives. In flood protection, those measures that are least invasive to the natural environment should be used first; in particular, non-technical flood protection methods (e.g., flood risk assessment and management, and the proper definition and management of flood plains). One of the bases for the sustainable development of water management is the preparation of models that can help us calculate the likelihood of maximum flow and to identify areas that are at risk of flooding. On this basis, the proper spatial policy and prevention of flood effects will be possible. This article presents the probabilistic analysis carried out on the flood risk dynamics for a selected area of the Odra River basin. The authors based their risk dynamic assessment on the results from the distributions of the maximum values for a selected hydrological characteristic – the flow rate. Based on the daily flow data from the years of 1994–2013 collected at a hydrological station on the Odra River in Malczyce, the 30-day flow maxima were set individually for four 5-year periods. Then, a probabilistic model of the maximum flow was developed based on these peaks for each 5-year period. The resulting models were used to estimate flood risks and for analyzing the dynamics for the studied area.
EN
We consider matrix Sturm-Liouville operators generated by the formal expression [formula] in the space [formula]. Let the matrix functions P := P(x), Q := Q(x) and R := R(x) of order n (n ∈ N) be defined on I, P is a nondegenerate matrix, P and Q are Hermitian matrices for x ∈ I and the entries of the matrix functions[formula], Q and R are measurable on I and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices P, Q and R that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by [formula]. In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.
4
Content available A two cones support theorem
EN
We show that if the Radon transform of a distribution ƒ vanishes outside of an acute cone Co, the support of the distribution is contained in the union of Co and another acute cone C1, the cones are in a suitable position, and ƒ vanishes distributionally in the direction of the axis of C1, then actually supp ƒ C ⊂ Co. We show by examples that this result is sharp.
EN
Distribution optimization of elastic material under elastic isotropic rectangular thin plate subjected to concentrated moving load is investigated in the present paper. The aim of optimization is to damp its vibrations in finite (fixed) time. Accepting Kirchhoff hypothesis with respect to the plate and Winkler hypothesis with respect to the base, the mathematical model of the problem is constructed as two–dimensional bilinear equation, i.e. linear in state and control function. The maximal quantity of the base material is taken as optimality criterion to be minimized. The Fourier distributional transform and the Bubnov–Galerkin procedures are used to reduce the problem to integral equality type constraints. The explicit solution in terms of two–dimensional Heaviside‘s function is obtained, describing piecewise–continuous distribution of the material. The determination of the switching points is reduced to a problem of nonlinear programming. Data from numerical analysis are presented.
PL
Bazując na dwóch znanych reprezentacjach funkcji Diraca o argumencie rzeczywistym w pracy zaproponowano jej uogólnienie na przypadek argumentu zespolonego. Na tej podstawie uzyskano szereg wzorów całkowych mogących znaleźć zastosowanie w rozwiązywaniu zagadnień fizyki matematycznej. Zaprezentowano przykład zastosowania funkcji delta Diraca o argumencie zespolonym w rozwiązywaniu zagadnienia rozpraszania fali elektromagnetycznej na narożu przewodzącym.
EN
In the paper, based on two known Dirac function representations with real argument they generalization for complex argument was proposed. Following, the set of integral formulas was derived, which could be applied for solving various problems as mathematical physics. As illustration an example of application of Dirac function with complex argument for solving problem of an electromagnetic wave scattering on a conductive corner was presented. (Dirac function with complex argument and example of its application in electromagnetism).
EN
The paper presents an idea to use weak derivatives in interval global optimization. It allows using the Newton operator to narrow domains of non-differentiable functions. Preliminary computational experiments are also presented.
EN
We define a semi-symmetric non-metric connection in a nearly Sasakian manifold and we consider semi-invariant submanifolds of a nearly Sasakian manifold endowed with a semi-symmetric non-metric connection. Moreover, we also obtain integrability conditions of the distributions on semi-invariant submanifolds.
9
Content available remote Division of distributions by locally definable quasianalytic functions
EN
We demonstrate that the Łojasiewicz theorem on the division of distributions by analytic functions carries over to the case of division by quasianalytic functions locally definable in an arbitrary polynomially bounded, o-minimal structure which admits smooth cell decomposition. Hence, in particular, the principal ideal generated by a locally definable quasianalytic function is closed in the Frechet space of smooth functions.
EN
The distributions or generalized functions are linear and continuous functionals defined by the class of functions which become null outside of a compact set and have derivatives of any order. The calculus with distributions was used to the modeling of linear systems. Generalized functions are also useful in the study of non-linear systems. In this paper, it is proved that the distributions with compact support represent a first approximation in the mathematical modeling of a system with infinite fading memory. The demonstration of this statement is the main part of the paper. The mathematical tool used is the differential calculus in the locally-convex topological space of the histories of inputs in system. The last part refers to the ε-distribution, R. Vallée's recent concept, and enumerates some applications.
11
EN
We review some results concerning multiplication of distributions using a pedagogical approach: we will first sketch the problems which may arise trying to multiply two distributions and. after discussing some mathematical tools, we will introduce several definitions of multiplication. In particular we give the details of a recent definition which works in any spatial dimension. As a particular application, we prove that delta functions and their derivatives can be multiplied.
PL
W artykule przedstawiono podejście dydaktyczne do zagadnienia mnożenia dystrybucji. Na początku pokazano na poziomie podstawowym, jakie problemy pojawiają się. gdy mnożone są dystrybucje. Następnie wprowadzono zaawansowane narzędzia matematyczne do analizy problemu i podano szereg użytecznych definicji mnożenia dystrybucji, które mogą mieć zastosowanie w naukach inżynieryjnych. W szczególności przeanalizowano dogłębnie możliwości jednej z zaproponowanych definicji dla mnożenia dystrybucji i jej pochodnych w przestrzeniach o dowolnym wymiarze.
EN
We show that in a sense of distributions [formula], where δ is the Dirac distribution, τ, x ∈ R and Kν(x) is the modified Bessel function. The convergence is in E'(R) for any even varphi(x) ∈ E(R) being a restriction to R of a function varphi(z) analytic in a horizontal open strip Ga = {z ∈ C: |Im z| < a, a > 0} and continuous in the strip closure. Moreover, it satisfles the condition [formula], |Re z| → ∞, α > 1 uniformly in ‾Ga. The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions.
EN
Results on products of Schwartz distributions are obtained when they have coinciding point singularities and only sums of the products exist in the distribution space. These results follow the pattern of a well-know distributional product published by Jan Mikusiński in 1966, and are named Mikusiński type products. The formulas are derived as the distributions are embedded in Colombeau algebra of generalized functions. This algebra possesses optimal properties regarding the distributional multiplication, and its notion of "association" allows one to obtain the results in terms of distributions.
14
Content available remote Geodesics in the sub-Lorentzian geometry
EN
The aim of this paper is to introduce the notion of sub-Lorentzian manifolds (which is done by analogy to sub-Riemannian manifolds) and to describe basic properties of such manifolds. In particular, we investigate problems related to the existence of the longest curves between two given points, and examine some conditions for continuity and differentiability of the (local) sub-Lorentzian distance function.
15
Content available remote Differential properties of the sub-Riemannian distance function
EN
Let (M, H, g) be a sub-Riemannian manifold. Fix a point [p_0 belongs to M] and denote by f the sub-Riemannian distance from [p_0]. It is proved that f is smooth on an open and dense subset of a certain neighbourhood of a regular geodesic. On the other hand, each minimizing geodesic around which f is smooth is regular.
EN
Using model assemblages generated by a FORTRAN program the parameter values of the slope of the power function and the factor of the exponential model of species-area relationships have been studied. It appeared that the slope value is not a constant independent of area and sampling method but depends strongly on grain, sampling method and model fit. The fraction of singletons in the sample proofed to be of major importance. A plot of slope against assemblage structure (estimated by the standard deviation of log2 (densities) was bell shaped with the highest slope values at intermediate SD values. A comparison of this plot with SD values from theoretical relative abundance distributions showed that log-normal distributed assemblages should have slope values that are higher than previously reported in the literature. Although it was impossible to predict the slope from the relative abundance distribution, the opposite was possible. At any given slope value there are two linked relative abundance distributions. The factor of the exponential model was more independent of sampling methods but linearily connected with sampling efficacy. A high non-linear correlation between factor and Shannon diversity was detected and a general function of this relationship developed and tested. The factor of the exponential species-area relationship may serve as an estimate of regional diversity.
17
Content available remote Water tree lengths and their distributions
EN
No formal approach has been developedso far for statistical analysis of water tree lengths. The methods usually employedcan be improvedby using cumulathe distribution plots. In thispaper, Weibull and log-normal plots were studied for this purpose. It was shown graphically that experimental measurement data samplesfit better on a log-normal plot in comparison with Weibull. The truncated nature of water tree measurement data does not allow a quantitative statistical inference to discriminate between the two populations. A Monte Carlo simulation was devised to resolve this problem. Simulation results show that if samples are truncated their distributions fit better in the log-normal model even if the sample originates from a Weibullpopulation. Nonetheless, further analysis of the simulation results proves that the nature of the population of water tree lengths is most likely to be log-normal.
PL
Nie istnieje jak dotąd formalne podejście do zagadnienia statystycznej analizy długości drzewień wodnych. Stosowane dotąd metody nie biorą na ogół użytku z wykresów dystrybuanty empirycznej, co mogłoby poprawić wnioskowanie. Rozkład log-normalny i rozkład Weibulla rozważane są w tym artykule jako odpowiednie do tego celu. Uzyskane wykresy pokazują graficznie, że dystrybuanty empiryczne wydają się mieć lepszą zgodność z rozkładem log-normalnym w porównaniu z rozkładem Weibulla. Ze względu na niepełny rodzaj próbki pochodzącej z pomiarów długości drzewień wodnych bardziej obiektywne statystyczne testy zgodności rozkładów nie mogły być tu zastosowane. By rozstrzygnąć ten problem użyto metodę Monte Carlo do symulacji pomiarów. Wyniki symulacji wykazały, że w wypadku próbki danych uciętych z lewa, takich jak w przypadku pomiarów drzewień wodnych, dystrybuanty empiryczne wydają się zgodne z modelem log-normalnym nawet jeżeli próbka pochodzi z populacji o rozładzie Weibulla. Niezależnie od tej obserwacji, dalsza analiza wyników symulacji przeprowadzona w artykule prowadzi do wniosku, że rozkład statystyczny całej populacji długości drzewień wodnych jest najprawdopodobniej log-normalny.
18
Content available remote Vibrations of an Elastic Body Carrying a Number of Concentrated Masses
EN
The paper investigates the existence and uniqueness of the weak solution to the initial boundary value problem for a system consisting of a hyperbolic-type partial differential equation with distributional coefficients and a collection of ordinary differential equations, modelling small vibrations of a mechanical system composed with an elastic body and a number of concentrated masses.
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