Multistep numerical methods for functional-differential-algebraic equations

This paper deals with constructing multistep numerical methods for differential delay equations under additional algebraic constraints. Theorems on the orders of convergence of these methods are proved both for functional-differential equations with algebraic constraints and for singular functional-differential equations.     

The double exponential map and covariant derivation

We study the double exponential map which is a composition of a special form of two exponential maps on a manifold with connection. We relate this map and the composition of covariant derivations as well as the composition of pseudodifferential operators on these manifolds.     

On the conditions under which a satellite orbit intersects the surface of a central body of finite radius in the restricted double-averaged three-body problem

This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M.L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body. Original Russian Text ? V.I. Prokhorenko, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 156–173.     

Ultrashort pulse laser osteotomy

Laser treatment of bone tissue has already been the subject of many studies to find substitutes for mechanical instruments that are nowadays used in implantology and orthopaedics. Mainly lasers with pulse durations in the μs region have been tested leaving surface features that are not always satisfactory. Therefore, in the present study laser osteotomy has been performed with a 330 fs Yb:glass laser, λ = 1040 nm, at a pulse repetition rate of 1 kHz. For bovine spongiosa, compacta, and cartilage, the ablation thresholds as well as the ablation rates for various pulse energies have been determined. Additionally, quadratic areas have been ablated in bone tissue. The remaining morphology has been analyzed via scanning electron microscopy. Laser ablation has also been performed with an Er, Cr:YSGG laser, λ = 2780 nm, PRR = 20 Hz, τ ～ 50 μs. The results of USPL and Erbium laser ablation are compared.     

A method for automatic correction of drift-distorted SPM images

A method for automatic correction of drift-distorted surface images obtained using a scanning probe microscope is developed. For the case of slowly varying drifts, simple linear transformations describing the observed distortions are constructed. One or two pairs of counter-scanned images (CSIs) are used as initial data. A system of linear equations is solved by finding the coordinate of the same surface feature in each CSI. The solutions obtained, i.e., linear transformation coefficients, are used to correct drift-induced distortions in horizontal and vertical planes. Two nonlinear drift correction methods are proposed, which provide a better accuracy in comparison with the linear correction method. The developed method can be used in feature-oriented scanning of the surface.     

Neutron reflection from a helical system

Analytical expressions for neutron reflection and transmission by magnetic mirrors with helical magnetization are derived. Reflectance curves with and without neutron spin flip are presented. Resonant properties of helical systems are studied.     

Carleman problem for two pairs of functions in a ring

We consider a particular case of the matrix Carleman problem for two pairs of functions in a ring and find a constructive solution of this problem. In addition, we propose an algorithm for the construction of solutions for two infinite systems of smooth transition and for a system of two singular equations of special type. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 662–671, May, 1997     

On the embedding problem with noncommutative kernel of order p4. VI

In the case of number fields the embedding problem of a p-extension with non-Abelian kernel of order p⁴ is studied. The two kernels of order 3⁴ with generators α, γ and relations α⁹ = 1, [α,α]³=1,[α,αγγ]==1,[αγγ]=α³,γ³=1 or γ³=α³ and the kernel of order 2⁴ with generators α, β, γ and relations α⁴=1 β²,[αγ]=1, [α,γ]=1,[βγ]=α² are considered. For kernels of odd order the embedding problem is always solvable. For the kernel of order 16 the solvability conditions are reduced to those for the associated problems at the Archimedean points, and to the compatibility condition. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 74–82.