1,677 research outputs found
Numerical computation of real or complex elliptic integrals
Algorithms for numerical computation of symmetric elliptic integrals of all
three kinds are improved in several ways and extended to complex values of the
variables (with some restrictions in the case of the integral of the third
kind). Numerical check values, consistency checks, and relations to Legendre's
integrals and Bulirsch's integrals are included
Approximation by planar elastic curves
We give an algorithm for approximating a given plane curve segment by a
planar elastic curve. The method depends on an analytic representation of the
space of elastic curve segments, together with a geometric method for obtaining
a good initial guess for the approximating curve. A gradient-driven
optimization is then used to find the approximating elastic curve.Comment: 18 pages, 10 figures. Version2: new section 5 added (conclusions and
discussions
Algebraic transformations of Gauss hypergeometric functions
This article gives a classification scheme of algebraic transformations of
Gauss hypergeometric functions, or pull-back transformations between
hypergeometric differential equations. The classification recovers the
classical transformations of degree 2, 3, 4, 6, and finds other transformations
of some special classes of the Gauss hypergeometric function. The other
transformations are considered more thoroughly in a series of supplementing
articles.Comment: 29 pages; 3 tables; Uniqueness claims and Remark 7.1 clarified by
footnotes; formulas (28), (29) correcte
Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT
We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions over the domain . Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to -convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem
Redundant Picard-Fuchs system for Abelian integrals
We derive an explicit system of Picard-Fuchs differential equations satisfied
by Abelian integrals of monomial forms and majorize its coefficients. A
peculiar feature of this construction is that the system admitting such
explicit majorants, appears only in dimension approximately two times greater
than the standard Picard-Fuchs system.
The result is used to obtain a partial solution to the tangential Hilbert
16th problem. We establish upper bounds for the number of zeros of arbitrary
Abelian integrals on a positive distance from the critical locus. Under the
additional assumption that the critical values of the Hamiltonian are distant
from each other (after a proper normalization), we were able to majorize the
number of all (real and complex) zeros.
In the second part of the paper an equivariant formulation of the above
problem is discussed and relationships between spread of critical values and
non-homogeneity of uni- and bivariate complex polynomials are studied.Comment: 31 page, LaTeX2e (amsart
Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity
We find two series expansions for Legendre's second incomplete elliptic
integral in terms of recursively computed elementary functions.
Both expansions converge at every point of the unit square in the plane. Partial sums of the proposed expansions form a sequence of
approximations to which are asymptotic when and/or
tend to unity, including when both approach the logarithmic singularity
from any direction. Explicit two-sided error bounds are given at
each approximation order. These bounds yield a sequence of increasingly precise
asymptotically correct two-sided inequalities for . For the
reader's convenience we further present explicit expressions for low-order
approximations and numerical examples to illustrate their accuracy. Our
derivations are based on series rearrangements, hypergeometric summation
algorithms and extensive use of the properties of the generalized
hypergeometric functions including some recent inequalities.Comment: 26 pages, 1 figures, 3 tables with numerical experiment
- …