Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.55 (2023)

In this paper, we study the integrability problem of a mathematical models of forests with two age classes of the form, ẋ =ρy − (y − 1)²x − sx, ẏ = x − hy, where ρ, h, s ∈ R. We proved that the system has a unique Darboux polynomial if and only if ρ = 0. The model has only two or three exponential factors if h ≠ 0 or h = 0, respectively. It is also, showed that the system admits a Darboux first integral if and only if ρ = h = 0 and has no analytic first integral in any neighborhood of fixed point except when ρ = h = 0.

We introduce a class of one-ended open 3-manifolds which can be ‘recursively’ defined from two compact 3-manifolds, and construct examples of manifolds in this class which fail to have a toric decomposition in the sense of Jaco-Shalen and Johannson.

n this paper we have obtained some novel generating functions of Fn(α, β−α; x) – a modified form of the extended Jacobi polynomial Fn(α, β; x) – by means of Weisner’s group-theoretic method with the suitable interpretation of the parameter α of the polynomial under consideration. Moreover, we have shown that the generating functions involving the extended Jacobi polynomial Fn(α, β; x) derived in [2], obtained by using the same Weisner’s group-theoretic method with the suitable interpretation of (α, β), can be easily obtained from our results. Finally, a group-theoretic method of obtaining general bilateral generating relation from general unilateral generating relation is also discussed.

In this paper we prove that the space of circle expanding maps of degree 2 preserving Lebesgue measure is an arc-connected space homeomorphic to an infinite-dimensional Lie group whose fundamental group is Z. The techniques involved in the proof are rather unexpected and lead to a formulation of a conjecture generalizing this result to higher dimensional infra-nilmanifolds.

We advertise elementary symmetric polynomials ei as the natural basis for generating series Ag,n of intersection numbers of ψ-classes on the moduli space of stable curves of genus g with n marked points. Closed formulae for Ag,n are known for genera 0 and 1 — this approach provides formulae for g = 2, 3, 4, together with an algorithm to compute the formula for any g. The claimed naturality of the ei basis relies in the unexpected vanishing of some coefficients with a clear pattern. As an application of the conjecture, we find new integral representations of Ag,n, which recover expressions for the Weil-Petersson volumes in terms of Bessel functions.

We give explicit formulas for the Berezin symbols and the complex Weyl symbols of the metaplectic representation operators by using the holomorphic representations of the Jacobi group. Then we recover some known formulas for the symbols of the metaplectic operators in the classical Weyl calculus, in particular for the classical Weyl symbol of the exponential of an operator whose Weyl symbol is a quadratic form.

The main objective of the paper is to give the specific forms of the meromorphic solutions of the equation $f^{n}(z)f(z+c)+P_{d}(z,f)=p_{1}(z)e^{\alpha_{1}(z)}+p_{2}(z)e^{\alpha_{2}(z)}$, where $c\in \mathbb{C}\setminus\{0\}$, $P_d(z,f)$ is a differential-difference polynomial in $f$ of degree $d\leq n-1$ with small functions of $f$ as its coefficients, $p_1, p_2(\not\equiv 0)$ are rational functions and $\alpha_1$, $\alpha_2$ are non-constant polynomials.

This paper is concerned with a system of magnetic effected piezoelectric beams with distributed delay term, where the heat flux is given by Cattaneo’s law (second sound). We prove the existence and the uniqueness of the solution using the semigroup theory. Then, we establish the exponential stability of the solution by introducing a suitable Lyapunov functional.

We give explicit formulas for the Berezin symbols and the complex Weyl symbols of the metaplectic representation operators by using the holomorphic representations of the Jacobi group. Then we recover some known formulas for the symbols of the metaplectic operators in the classical Weyl calculus, in particular for the classical Weyl symbol of the exponential of an operator whose Weyl symbol is a quadratic form.