A constructive method for uniform approximation by means of integrated Lagrange interpolation polynomials in the spaceC 1[a,b]

Пусть интерполяционные уз лыa <x ₁ <x ₂ < ... <х _n <b являются корнямиn-го полинома системы полиномов {w _n(x)} _n ^∞ =1, ортогонально й с весомρ(х), 0 < ρ(х) ∈L[a, b], причем 1/ρ(x) ∈L[a, b]. Доказано, что еслиf(х) ∈С ¹ [a, b], то проинтегрированн ые полиномы $$Q_n (x) = \smallint _a^x L_{n - 1} (f',t)dt + f(a)$$ интерполяционных по линомов Лагранжа $$L_{n - 1} (f,x) = \sum\limits_{k = 1}^n {f'(x_k )\frac{{\omega _n (x)}}{{\omega '_n (x_k )(x - x_k )}}}$$ порядкаn?1 функцииf′(x) сходятся равномерно к функцииf(х) в замкнуто м интервале [а,b]. Во второй части стать и, где вместо условия 1/ρ(х) ∈L[a, b] предполагается, чтоρ(х) ∈С ²[а, b],ρ(х) > 0,ρ′ (х) > 0, доказываются подобные утверждени я для проинтегрированных интерполяционных по линомов Лагранжа и для прибли жений рациональными дробя ми. В работе даются неулу чшаемые порядковые о ценки и описываются примене ния полученных результа тов в теории дифферен циальных уравнений.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Fractal dimension,attractors, and the boussinesq approximation in three dimensions

The Boussinesq approximation, where the viscosity depends polynomially on the shear rate, finds more and more frequent use in geological practice. In the paper, this modified Boussinesq approximation is investigated as a dynamical system for which the existence of a global attractor is proved. Finally, a new criterion for estimating the fractal dimension of invariant sets is formulated and its application to the problem under consideration is illustrated.     

On approximation by bivariate incomplete polynomials

In this paper we prove that given certain convex domains Δ on the plane, ε>0, andf∈C(Δ) such thatf=0 on θ²Δ={(θ² x,θ² y):(x,y)?Δ} (0<θ<1), a polynomialp(x, y) of the form $$p(x,y) = \sum\limits_{\theta n \leqslant k + l \leqslant n} {a_{kl} x^k y^l }$$ exists such that ∥f?p∥ _C(Δ) ≤ε. The admissible convex domains include triangles and parallelograms with a vertex at the origin and sections of unit disk.     

A weak approach to finite elasticity

On the ground of four axioms we define thekinematics of perfectly elastic bodies and in particular the notion ofweak deformations of a perfectly elastic body. Weak deformations turn out to agree withweak diffeomorphisms introduced in [10], a class of rectifiable currents which enjoys good closure and compactness properties. Defining thedynamics of perfectly elastic bodies in terms of twoconstitutive conditions on the stored energy function, we can therefore prove existence of stable equilibrium weak deformations for mixed boundary value problems, which moreover satisfy equilibrium and conservation equations.This work has been partially supported by the Ministero dell'Universitá e della Ricerca Scientifica, by C.N.R. contract n. 91.01343.CT01, by the European Research Project GADGET, and by Grant n.11957 of the zech Acad. of Sciences.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

Bell-type inequalities in horizontal sums of Boolean algebras

We give a necessary and sufficient condition for a Bell-type inequality to hold in a horizontal sum of finitely many finite Boolean algebras.     

Difference posets,effects, and quantum measurements

Difference posets as generalizations of quantum logics, orthoalgebras, and effects are studied. Observables and measures generalizing normalized POV-measures and generalized measures on sets of effects are introduced. Characterization of orthomodularity of subsets of a difference poset in terms of triangle closedness and regularity of these subsets enables us to characterize observables with a Boolean range. Boolean powers of difference posets are investigated; they have similar properties to that of tensor products, and their connection with quantum measurements is studied.     

Oxygen incorporation in polyethylene implanted with 150 keV Sb+ ions

Samples of polyethylene (PE) loaded with carbon black up to 8 wt.% and implanted with 150 keV Sb⁺ ions to the doses from 2×10¹³–2×10¹⁵ cm^–2 were studied using standard Rutherford Back Scattering (RBS) technique. On the PE samples implanted to the doses above 2×10¹⁴ cm^–2, a considerable surface carbonization is observed. The measured parameters of the Sb depth profile are compared with theoretical TRIM estimations. The projected range is by 25% lower than the theoretical one and the range straggling is about twice of that predicted. The differences are explained by stepwise polymer degradation during the ion bombardment. Strong oxidation of the ion implanted polymers is also observed. The oxygen depth profiles from the sample surface up to the depth comparable with Sb⁺ ion range evolve from nearly uniform one for low ion doses to highly non-uniform one for doses above 1×10¹⁵ cm^–2. The total oxygen content in the sample surface layer 300 nm thick reaches a maximum for the doses of (1–2)×10¹⁴ cm^–2.On leave from Belorussian State University, 220050 Minsk, Belarus.The authors thank the member of electrostatic accelerator group at INP for help during RBS measurements. The work was partly (V.H. and J.K.) supported by the Internal Grant Agency of the Academy of Sciences of Czech Republic under the grant No. 14805 and by the Internal Grant Agency of the Ministry of Education of Czech Republic under the grant No. 1002 (V.., V.R., V.H. and J.K.). In the final stage, the work was also supported by the Grant Agency of Czech Republic under the grant No. 202/93/0121.