We study Tauberian conditions for the existence of Cesàro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber’s second theorem on the converse of Abel’s theorem. For Schwartz distributions, we obtain extensions of many classical ...

0. Preface. This article is a survey of certain aspects of the theory of multiple Fourier and trigonometric series. I t is by no means meant to be a complete survey; for example, it is practically disjoint with the material covered on the subject in Zygmund's book [38, Chapter 17.]. There are eight sections to this survey. §1 is the introduction. §2, §3, and §4 are expository in the sense that ...

We conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam’s results for the graded Hecke algebra, respectively. These Paley– Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrary smooth functions. Furtherm...

In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including ...

Two classical problems on plane polynomial vector fields, Hilbert’s 16th problem about the maximal number of limit cycles in such a system and Poincaré’s center-focus problem about conditions for all trajectories around a critical point to be closed, can be naturally reformulated for the Abel differential equation y′ = p(x)y + q(x)y. Recently, the center conditions for the Abel equation have be...

The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjectur...

We discuss an Abel-Jacobi invariant for algebraic cobordism cycles whose image in topological cobordism vanishes. The existence of this invariant follows by abstract arguments from the construction of Hodge filtered cohomology theories in joint work of Michael J. Hopkins and the author. In this paper, we give a concrete description of the Abel-Jacobi map and Hodge filtered cohomology groups for...