In the paper two new approaches for construction of parametric polynomial approximants of a unit circle are presented. The obtained approximants have better approximation properties than those given ...by other methods, i.e., smaller radial error, symmetry, and exponential error decay.
•Two new approaches for construction of parametric polynomial approximants of a unit circle are presented.•The approximants have good approximation properties: small radial error, symmetry, exponential error decay.
In this paper a relation between graph distance matrices of the star graph and its generalizations and Euclidean distance matrices is considered. It is proven that distance matrices of certain ...families of graphs are circum Euclidean. Their spectrum and generating points are given in a closed form.
In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter ...that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too.
In this paper, Hermite interpolation by parametric spline surfaces on triangulations is considered. The splines interpolate points, the corresponding tangent planes and normal curvature forms at ...domain vertices and approximate tangent planes at midpoints of domain edges. Two variations of the scheme are studied: C1 quintic and G1 octic. The latter is of higher polynomial degree but can approximate surfaces of arbitrary topology. The construction of the approximant is local and fast. Some numerical examples of surface approximation are presented.
•Hermite surface approximation scheme based on Argyris element.•C1 scheme variation uses quintic and G1 variant uses octic triangular patches.•Interpolation of geometric data (points, tangent planes, normal curvature forms).•Local construction and linear complexity.
In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of ...the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain
G
1 cubic interpolatory spline.
It is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct. Thus the corresponding interpolation matrix M is nonsingular. ...Schumaker stated the conjecture that all principal submatrices of M are nonsingular too. Furthermore, all of the corresponding determinants (the principal minors) are conjectured to be positive. This result would solve the constrained interpolation problem. In this paper, the conjecture on minors for polynomial degree ⩽17 and conjecture for some particular configurations of domain points are confirmed.
A matrix is a Euclidean distance matrix (EDM) if there exist points such that the matrix elements are squares of distances between the corresponding points. The inverse eigenvalue problem (IEP) is as ...follows: construct (or prove the existence of) a matrix with particular properties and a given spectrum. It is well known that the IEP for EDMs of size 3 has a solution. In this paper all solutions of the problem are given and their relation with geometry is studied. A possible extension to larger EDMs is tackled.
A technique for determining the permeability of a phospholipid membrane on a single giant unilamellar vesicle (GUV) is described, which complements the existing methods utilizing either a planar ...black lipid membrane or sub-micrometre-sized liposomes. A single GUV is transferred using a micropipette from a solution of a nonpermeable solute into an iso-osmolar solution of a solute with a higher membrane permeability. Osmotical swelling of the vesicle is monitored with a CCD camera mounted on a phase contrast microscope, and a sequence of images is obtained. On each image, the points on the vesicle contour are determined using Sobel filtering with adaptive binarization threshold, and from these, the vesicle radius is calculated with great accuracy. From the time dependence of the vesicle radius, the membrane permeability is obtained. Using a test set of data, the method provided a consistent estimate of the POPC membrane permeability for glycerol, P = 1.7 x 10-8 m s-1, with individual samples ranging from 1.61 x 10-8 m s-1 to 1.98 x 10-8 m s-1. This value is -40% lower than the one obtained on similar systems. Possible causes for this discrepancy are discussed.
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