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1
Content available remote The real and complex convexity
EN
We prove that the holomorphic differential equation ϕ’’(ϕ+c) = γ(ϕ’)² (ϕ:C→C be a holomorphic function and (γ, c) ϵ C²) plays a classical role on many problems of real and complex convexity. The condition exactly γ ϵ [wzór] (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A₁, A₂) ϵ C² and g₁, g₂ : Cᵑ → C be two analytic functions. Put u(z, w) = │A ₁w - g₁(z) │² + │A₂w - g₂(z) │²v(z,w) = │A₁w - g₁(z) │² + │ A₂w - g₂(z) │², for (z,w) ϵ Cᵑ x C. We prove that u is strictly plurisubharmonic and convex on Cᵑ x C if and only if n = 1, (A₁, A₂) ϵ C² \{0} and the functions g₁ and g₂ have a classical representation form described in the present paper. Now v is convex and strictly psh on Cᵑ x C if and only if (A₁, A₂) ϵ C² \{0}, n ϵ {1,2} and and g₁, g₂ have several representations investigated in this paper.
2
Content available remote Location of zeros of Lacunary-type polynomials
EN
In this paper, we present some interesting results concerning the location of zeros of Lacunary-type of polynomial in the complex plane. By relaxing the hypothesis and putting less restrictive conditions on the coefficients of the polynomial, our results generalize and refines some classical results.
EN
The aim of this study was to investigate the curve fitting and model selection problem of the torque–velocity relationship of elbow flexors and extensors in untrained females. The second goal was to determine the optimal models in different function classes and the best, among the optimal ones. Lastly, test the best models to predict the torque were tested. Methods: Using the polynomials (second – fourth degree) and Boltzmann sigmoid functions, and a different presentation of data points (averages, a point cloud, etc.), we determined the optimal models by both error criteria: minimum residual sum of squares and minimum of the maximal absolute residue. To assess the best models, we applied Akaike and Bayesian information criteria, Hausdorff distance and the minimum of the smallest maximal absolute residue and the predictive torque–velocity relationships of the best models with torque values, calculated beyond the experimental velocity interval. Results: The application of different error and model selection criteria showed that the best models in the majority of cases were polynomials of fourth degree, with some exceptions from second and third degree. The criteria values for the optimal Boltzmann sigmoids were very close to those of the best polynomial models. However, the predicted torque–velocity relationships had physiological behavior only in Boltzmann’s sigmoid functions, and their parameters had a clear interpretation. Conclusion: The results obtained suggest that the Boltzmann sigmoid functions are suitable for modeling and predicting of the torque–velocity relationship of elbow flexors and extensors in untrained females, as compared to polynomials, and their curves are physiologically relevant.
EN
We give a new proof and discuss an extension of Jack’s lemma for polynomials.
EN
The main purpose of this paper is to study the controllability of solutions of the differential equation [...] In fact, we study the growth and oscillation of higher order differential polynomial with meromorphic coefficients in the unit disc [...] generated by solutions of the above kth order differential equation.
6
Content available remote On circularly symmetric functions
EN
[...] In this paper besides X we also consider some of its subclasses: X(...) and Y (...) S consisting of functions in X with the second coeffcient fixed and univalent starlike functions respectively. According to the suggestion, in Abstract we add one more paragraph at the end of the section: For X(...) we find the radii of starlikeness, starlikeness of order (alfa), univalence and local univalence. We also obtain some distortion results. For Y (...) S we discuss some coecient problems, among others the Fekete-Szego ineqalities.
EN
In this paper we present the modular computing structures (MCS) defined on the set of polynomials over finite rings of integers. This article is a continuation of research on the development of modular number systems (MNS) on arbitrary mathematical structures such as finite groups, rings and Galois fields [1-7].
8
Content available remote Zastosowanie metod aproksymacyjnych do optymalizacji składu mieszanek gumowych
PL
Wiele problemów technologicznych pojawiających się w przemyśle gumowym można rozwiązać w lepszy niż dotychczas sposób stosując aproksymacyjne metody optymalizacji. W artykule podano ogólne informacje dotyczące tych metod oraz przedstawiono 3 przykłady ich praktycznego zastosowania do optymalizacji składu mieszanek kauczukowych. Nie jest to jedyny sposób wykorzystania metod aproksymacyjnych. Można je z łatwością zastosować także do optymalizacji parametrów procesów technologicznych. Wyniki aproksymacji wizualizowano za pomocą wykresów konturowych, a do ustalania końcowych wyników optymalizacji posługiwano się także innymi modułami programu OPTY, służącymi m.in. do wyszukiwania składników mieszanki kauczukowej odpowiadających minimum współczynnika tarcia lub obszaru składników mieszanki, w których mieszanka kauczukowa ma właściwości mieszczące się w określonym przedziale.
EN
Many of the technical problems that appear in rubber industry can be resolved in better way than in the past if the approximation methods are applied. In this paper the general descriptions of approximation methods and three case studies of their use are presented. The chosen case studies concentrated on optimization of contents of rubber formulation, though this is not a sole aim of application of the approximation methods. Those methods can also easily be used to optimize parameters of the technological process as well. To aid visualization of the results, contour plots (contour line or isoline) have been used. The final results of optimization were determined using the additional modules of the elaborated software named OPTY, such as: finding the contents of the formulations of rubber compounds responding to the minimum of the coefficient of friction or the area of the contents of ingredients of rubber compounds in which rubber compound have the required properties.
PL
Jednym z najistotniejszych urządzeń wpływających na działalność i otoczenie człowieka jest silnik spalinowy, który niezmiennie stanowi podstawowe źródło napędu w transporcie drogowym. Do oceny właściwości silnika w całym zakresie wykonywanej pracy służą charakterystyki, czyli zależności parametrów roboczych od innego wskaźnika wpływającego na pracę jednostki napędowej.
10
Content available remote The polynomial tensor interpolation. Arithmetical case
EN
In this paper the tensor interpolation by polynomials of several variables is considered. The effective formulas for polynomial coefficients for arithmetical case were obtained.
11
Content available remote Generalizations of two-index two-variable Hermite matrix polynomials
EN
In this paper, we introduce a new generalization of the Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. An explicit representation and an expansion of the matrix exponential in a series of these matrix polynomials is obtained. Properties of Hermite matrix polynomials such as the recurrence formula permit an efficient computations of matrix functions are established. A new expansions of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.
12
Content available remote Derivation with engel conditions on multilinear polynomials in prime rings
EN
Let R be a prime ring with extended centroid C and characteristic different from 2, d a nonzero derivation of R, f(x1,..., xn) a nonzero multilinear polynomial over C such that [d2(f(x1,... , xn)),d(f(x1,... ,xn))]k = 0 for all x1,... ,xn in some nonzero right ideal [...] of R, where k is a fixed positive integer. If d(p) p [..]0, then pC = eRC for some idempotent e in the socle of RC and /(x1,..., xn) is central-valued on eRCe.
13
Content available remote The polynomial tensor interpolation
EN
In this paper the tensor interpolation by polynomials of several variables is considered.
14
Content available remote Polynomials in additive functions and generalized polynomials
EN
We consider polynomials P in additive functions g1,... , gm and present two approaches for a characterization of those generalized polynomials p, which may be represented as p = P o (g1,..., gm). Under rather general assumptions on the domains of the gi and of P, one of the approaches is based on certain properties of subspaces generated by translates of p. The other approach utilizes the fact, that every p is the diagonalization of an associated multi-Jensen function.
PL
W referacie zaprezentowano wyniki porównania niepewności przewidywanych wartości funkcji znalezionych na podstawie aproksymacji wyników pomiaru zwykłymi algebraicznymi oraz ortogonalnymi wielomianami Czebyszewa. Przedstawione są wzory analityczne do obliczenia niepewności tych funkcji.
EN
In the paper the comparison results of the uncertainty of the forecasted values of function, obtained as a result of the measurement result approximation by both usual algebraic and Chebyshev polynomials are presented. The obtained formulas for calculation uncertainty of these function are presented.
EN
The paper presents a powerful algorithm useful to solve the problem of experimental data values finding don't occur within the registered experimental data set by using the one-dimensional polynomial interpolation procedure via a cubic spline method. Described free algorithm is presented in the form of Matlab routine with short author's comments.
17
Content available remote Wave polynomials for solving different types of two-dimensional wave equations
EN
The paper demonstrates a specific power series expansion technique used to obtain the approximate solution of the two-dimensional wave equation in some unusual cases. The solution for inhomogeneous wave equation, for more complicated shape geometry of the body, discrete boundary conditions and a membrane whose thickness is not constant is shown. As solving functions (Trefftz functions), so-called wave polynomials are used. Recurrent formulas for the particular solution are obtained. Some examples are included.
EN
The author presents a proof of the theorem on a necessary and suficient condition for the semiscalar equivalency of the polynomial matrices with the relatively prime elementary divisors in terms of the Δ, C transformations of certain way builded numerical matrices.
EN
The paper presents new results related to explicit solutions of semi-discrete models which appear in studies of transients in lossy long transmission lines. The theory of nonlinear difference equations has been applied. It is shown that the ratio of transversal and longitudinal state variables distributed along the line can be easily determined by using power polynomials in indeterminate q = q(s) depending on the line parameters and the complex frequency s. The established solutions are expressed in terms of appropriate continued fractions and the corresponding generating matrices are involved. Problems concerning some equalities and the limiting behaviour as well as parameters determining initial terms are also studied. Conditions leading to strange attractors for semi-discrete models are formulated and studied. Basins of attraction of the strange attractors are exhibited.
PL
W artykule przedstawione zostały nowe wyniki odnoszące się do jawnych rozwiązań równań opisujących pół-dyskretne modele stosowane w badaniach stanów nieustalonych w liniach długich. Podstawę w zrealizowanych badaniach stanowi teoria nieliniowych równań różnicowych. Wykazane zostało, że stosunek transwersalnej i podłużnej zmiennych stanu rozłożonych wzdłuż linii może być stosunkowo łatwo wyznaczony przez zastosowanie wielomianów potęgowych względem zmiennej niezależnej q = q(s), która zależy od parametrów jednostkowych linii długiej i częstotliwości uogólnionej s. Podano reguły wyznaczania tych wielomianów oraz zależności rekurencyjne pozwalające określać współczynniki przy kolejnych potęgach zmiennej q. Przedstawiono też możliwość pojawiania się zjawiska chaosu w rozkładach napięć i prądów w linii długiej przy odpowiednim obciążeniu na krańcu odbiorczym i zasilaniu impulsowym na wejściu linii.
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