As Lie group theory has developed it has also become more and more pervasive in its influence on other mathematical disciplines. The original founder of this theory was a Norwegian, Marius Sophus Lie, who was born in Nordfjordeid, 1842. In order to understand the background and motivation to Lie's work we must go further back. A central problem in algebra at the end of the 18th century was that...

We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m, 1) to the corresponding p...

Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey’s 6ψ6 bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the 6ψ6 sum. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the q-Gosper algorithm. It is demo...

In this paper we present a method of studying convolution operator under the Sonin conditions imposed on kernel. The particular case kernel is fractional integral Riemman–Liouville operator, other various types kernels are Bessel-type function, functions with power-logarithmic singularities at origin e.t.c. We pay special attention to study close power type functions. main our aim Sonin–Abel eq...

A new method of obtaining exact solution for the rate matrix R in the Matrix-Analytic method in case of the phase state of dimension two is proposed. The method is based on symbolic solution of the determinental polynomial equation, and obtaining a linear matrix equation for the unknown rate matrix R by Cayley–Hamilton theorem. The method is applied to analyze the EnergyPerformance tradeoff of ...

We consider an Abel equation (∗)y′ = p(x)y2+q(x)y3 withp(x), q(x)—polynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is that y0 = y(0) ≡ y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients vk(1) in the development y(x) = y0 + ∑∞ k=2 vk(x)y k 0 . Follo...

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel’s type integral equation. Obtaining the physically relevant quantity from the measured one requires, therefore ...