Some enhancements to the approximation of one-variable functions with respect to an orthogonal basis are considered. A two-step approximation scheme is presented here. In the first step, a constant bias is extracted from the approximated function, while in the second, the function with extracted bias is approximated in a usual way. Later, these two components are added together. First of all we prove that a constant bias extracted from the function decreases the error. We demonstrate how to calculate that bias. Secondly, in a minor contribution, we show how to choose basis from a selected set of orthonormal functions to achieve minimum error. Finally we prove that loss of orthonormality due to truncation of the argument range of the basis functions does not effect the overall error of approximation and the expansion coefficients' correctness. We show how this feature can be used. Our attention is focused on Hermite orthonormal functions. An application of the obtained results to ECG data compression is presented.
Some stronger and equivalent metrics are defined on M, the set of all bounded normal operators on a Hilbert space H and then some topological properties of M are investigated.
Let L1 = −Δ + V be a Schrödinger operator and let L2 = (−Δ)2 + V2 be a Schrödinger type operator on \mathbb{R}^{n}\left ( n\geqslant 5 \right ) where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s\geqslant n/2. The Hardy type space H_{L2}^{1} is defined in terms of the maximal function with respect to the semigroup \left\{ {{e^{ - t{L_2}}}} \right\} and it is identical to the Hardy space H_{L2}^{1} established by Dziubański and Zienkiewicz. In this article, we prove the Lp-boundedness of the commutator Rb = bRf - R(bf) generated by the Riesz transform R = {\nabla ^2}L_2^{ - 1/2} , where b \in BM{O_\theta }(\varrho ) , which is larger than the space BMO\left (\mathbb{R}^{n} \right ). Moreover, we prove that Rb is bounded from the Hardy space H_{L2}^{1} into weak L_{weak}^1 (\mathbb{R}^n )., Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang., and Obsahuje seznam literatury
Let λ1(Q) be the first eigenvalue of the Sturm-Liouville problem y ′′ − Q(x)y + λy = 0, y(0) = y(1) = 0, 0 < x < 1. We give some estimates for mα,β,γ = inf Q∈Tα,β,γ λ1(Q) and Mα,β,γ = sup Q∈Tα,β,γ λ1(Q), where Tα,β,γ is the set of real-valued measurable on [0, 1] x α(1 − x) β -weighted Lγ-functions Q with non-negative values such that ∫ 1 0 x α(1 − x) βQ γ (x) dx = 1 (α, β, γ ∈ ℝ, γ ≠ 0).
We consider the Sturm-Liouville problem with symmetric boundary conditions and an integral condition. We estimate the first eigenvalue λ1 of this problem for different values of the parameters.
Some ethological aspects of the interrelations between ants and the larvae of Blasticotoma filiceti Klug, 1834 were investigated in the Altai Republic and Novosibirsk Region in 2006-2008. The interactions of ants with the larvae of this sawfly are determined by the concealed way of life style of B. filiceti. The majority of the ant-sawfly encounters occurred near holes in fern fronds at the moment when larvae excreted liquid or left their tunnels before descending to the soil prior to overwintering. Sawfly larvae visited by more aggressive ants, such as Formica s. str., leave the fern fronds slowly, which enables them to avoid inciting attacks by ants. The behaviour of the ants while collecting the larval excretion is similar to their behaviour at sugar troughs. The organisation of the collecting larval excreta by ants was investigated in detail in the cases of Formica polyctena Förster, 1850 and Myrmica rubra Linnaeus, 1758. The individual fern plants with sawfly larvae are attended by relatively constant groups of foragers in both cases. However, the highly social red wood ants interact with sawfly larvae in a more complex way. While the working groups of M. rubra tending sawfly larvae consist only of non-aggressive "unspecialized" foragers, those of F. polyctena include also a few "on duty" ants that protect the trophobionts, at least from the other ants. and Tatiana A. NOVGORODOVA, Olga B. BIRYUKOVA.
The appearance of isolated sunspot groups as well as clustering in large active regions depends upon a complex dynamo process. Evidence of this dynamo process may be deduced from three different outstanding studies. a) The emergence of new fluxes (spots or flares) is preceded by local cyclonic motions, observed at
the photospheric level. In regions of weak magnetic fields ; for example polarity inversion lines, gaps between magnetic "hills" or borders of the facula. This velocity structure is a response with a short scale in time and space to local subphotospheric perturbation and thus creates currents and new magnetic flux. b) Magnetic tracers such as long lived H filaments and sunspots, show that the regions of emergence of new flux (family of sunspot groups, eruptive sites or parasitic polaritics) are related to the existence of limited areas rotating rigidly. These "pivot points" which do not follow dlffcrential rotation, could be anchored more deeply than the active centers themselves. c) A large scale circulation, tied to the global rotation, reflects the motions of the underlying
fluid (frozen field). Recent results show the existence of azimuthal rolls which transport upward the deep magnetic field. They move slowly toward the poles, and they appear to govern the cyclicity
and to modulate the observed solar rotation. These observational results need to be considered to understand the production and the development of active regions.
We provide a characterization of continuous images of Radon-Nikodým compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.
This paper is a contribution to the theoretical foundations of data mining. More precisely, we contribute to a part of data mining allowing us to search for associations among attributes that can be expressed in the form of natural language sentences. The theoretical background and also a method for mining such associations was published recently in [V. Novák et al., Mining pure linguistic associations from numerical data, Int. Journal of Approximate Reasoning 48 (2008), 4 -- 22]. We elaborated other mathematical representations of the model presented in the mentioned paper in order to extend its applicability.
Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem., Ji-Cai Liu., and Seznam literatury