Smooth polynomial approximation of spiral arcs

Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bézier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bézier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.     

Harmonic type approximation lemmas

We give a survey of known and not known harmonic type approximation lemmas which are descendants of the classical De Giorgi's one, and we outline some of their recent or possible applications.     

Harmonic approximation of difference operators

For a general class of difference operators Hε=Tε+Vε on ?²(d(εZ)), where Vε is a multi-well potential and ε is a small parameter, we analyze the asymptotic behavior as ε→0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first n eigenvalues of Hε converge to the first n eigenvalues of the direct sum of harmonic oscillators on Rd located at the several wells. Our proof is microlocal.     

The problem of approximation in mean on arcs in the complex plane

Classical theorems on the approximation of curves in the complex domain are studied; in particular, direct and inverse theorems on the arcs Γ in the complex plane in the metric of L_p(Γ) are obtained. The results obtained are new in the case of a closed interval [?1, 1] as well.     

On the length preserving approximation of plane curves by circular arcs

A technique for the length preserving approximation of plane curves by two circular arcs is analyzed. The conditions under which this technique can be applied are extended, and certain consequences of the proved results unrelated to the approximation problem are discussed. More precisely, inequalities for the length of a convex spiral arc subject to the given boundary conditions are obtained. Conjectures on curve closeness conditions obtained using computer simulation are discussed.     

Extremal polynomials on arcs of the circle with zeros on these arcs

The paper gives a solution of an extremal problem of finding monic polynomial least deviating from zero on several arcs of the unit circle, under some restrictions on the location of zeros and additional conditions on mutual position of the arcs. The extremal polynomial is represented in the terms of density of harmonic measure. The work is done under the financial support of RFFI (Project 07-01-00167) and the President of RF Grant (Project NSh-2970.2008.1).     

Random arcs on the circle

In the paper one gives a short survey of the most important results of the author and of other statisticians regarding random coverings of the circle.Professor L. Holst (Uppsala University, Sweden) has given in December 1980 at the Leningrad Branch of the Mathematical Institute a survey lecture on this subject.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 108, pp. 188–192, 1981.     

Harmonic vibration of prismatic shells in zero approximation of Vekua's hierarchical models

In this article elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the zero approximation of I.Vekua's hierarchical models is considered. The well-posedness of the boundary value problems (BVPs) under the reasonable boundary conditions at the cusped edge and given displacements at the non-cusped edge is studied in the case of harmonic vibration. The approach works also for non-symmetric prismatic shells word for word. The classical and weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described and its connection with weighted Sobolev spaces is established. Moreover, some sufficient conditions for a linear functional arising in the right-hand side of the variational equation to be bounded are given.     

Families of arcs on rational surface singularities

We consider the spaceH of arcs on a rational surface singularity (S, P), with the proalgebraic structure induced by the truncation maps. We introduce some sets of arcs by imposing valuative conditions and we prove that they are closed subsets ofH. This leads to give a sufficient condition in order to have an affirmative answer for the problem of Nash. We conclude the solution of the problem for the minimal surface singularities. Supported by D.G.I.C.Y.T. PB91-0210-C02-01     

Maximal arcs and disjoint maximal arcs in projective planes of order 16

This paper tabulates the results of a number of computer searches in projective planes of order 16. Maximal arcs of degree 4 are found in all but two of the known planes of order 16 (and their duals). Any such arc yields a resolvable 2-(52, 4, 1) design that admits at least 52 resolutions. Pairs of disjoint degree 4 maximal arcs are also shown to exist in certain of the planes giving rise to 104-sets of type (4, 8).