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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 Partial sums of a certain harmonic univalent meromorphic functions
EN
In the present paper we determine sharp lower bounds of the real part of the ratios of harmonic univalent meromorphic functions to their sequences of partial sums [...].
3
Content available remote Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales
EN
Let (hk)k≥0 be the Haar system on [0,1]. We show that for any vectors ak from a separable Hilbert space H and any εk∈[−1,1], k=0,1,2,…, we have the sharp inequality ...[formula], where W([0,1]) is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ∥Y∥W(Ω)≤2∥X∥L∞(Ω), where X and Y stand for H-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.
EN
The paper presents a solution of motion planning and control of mobile robot in a two-dimensional environment with elliptical static obstacle based on hydrodynamics description. Theoretical background refers to solution of Laplace equation using complex algebra. The method of designing complex potential with respect to stationary elliptical obstacle and stationary goal is formally shown. Next, the planning motion problem is extended assuming that the goal is moving. Then results of motion planning is used in order to design closed-loop control algorithms which is based on decoupling technique. Theoretical considerations are supported by numerical simulations illustrating example results of motion planning and control.
EN
A recent result of Sibel Yelcin et al. [4] appeared in "Journal of Inequalities in Pure and Applied Mathematics" (2007) concerning the convolution of two harmonic univalent functions in the class [wzór] is improved.
6
Content available remote Generalized growth of entire harmonic functions
EN
Let H(x), x = (x1, x2, ... , xn), be an entire harmonic function in Rn.Fryant and Shankar [1] had obtained growth properties of H explicitly in terms of its Fourier coeffcients. In this paper, we obtain the characterizations of generalized order and type and introduce the generalized lower order for H. Special case of functions of slow growth has also been considered. Our results generalize some of the results obtained in [1].
7
Content available remote Sharp norm inequalities for martingales and their differential subordinates
EN
Suppose ƒ = (ƒn), g = (gn) are martingales with respect to the same filtration, satisfying |ƒn-ƒn-i| ≤ |gn -gn-1|, n = 1,2,..., with probability 1. Under some assumptions on ƒo, go and an additional condition that one of the processes is nonnegative, some sharp inequalities between the pth norms of ƒ and g, 0 < p < ∞, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.
PL
W artykule podjęto próbę opisu procesu mieszania czternastoskładnikowej paszy dla gołębi w mieszalniku z pionowym mieszadłem o działaniu okresowym. W wyniku przeprowadzonych pomiarów zaobserwowano charakter zależności zmiany procentowego udziału poszczególnych składników mieszaniny w funkcji czasu mieszania. Charakter tych zmian został opisany za pomocą funkcji drgań tłumionych. Do wymodelowania krzywej drgań tłumionych wykorzystano dwie metody: klasyczną oraz bayesowską.
EN
In the paper a description of the mixing process of a 14-component pigeon pellet in the periodical mixer with vertical agitator has been given. Following the performed measurements, a relation of the components' percentages in time has been observed. To model these changes, a damped vibration function was proposed. To estimate model parameters, a classical and a Bayesian approaches were adopted.
9
Content available remote Approximation of entire harmonic functions in R3 in Lß - norm
EN
A function which is harmonic in a neighbourhood of the origin in R(3) has there an expansion in spherical harmonics. For HK, he class of all harmonic functions regular in and continuous on DR , the closure of DK, we define approximation error as , is the set of all harmonic polynomials of degree at most n. In this paper we have obtained the necessary and sufficient condition on the rate of decrease of has an analytic continuation as an entire harmonic function with (p, q) - growth.
10
Content available remote A note on the Lie ball
EN
We give an effective formula for the maximal n-circled domain contained in the Lie ball.
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