Problems of Polarization

Political commentators across the spectrum point to polarization as the cause of dysfunctions that recently have beset democratic societies. But the concept of polarization is rarely examined. This chapter presents several distinct kinds of polarization, starting by distinguishing political polarization from belief polarization, and offers a view of how their interaction gives rise to political problems ranging from legislative deadlock and partisan animosity to escalating extremism. Although the aim of the chapter is largely diagnostic, it concludes with an indication of how polarization might be managed.

THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓp+ AND Lp−

The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

On Homoclinic Solutions for First-Order Superquadratic Hamiltonian Systems with Spectrum Point Zero

The existence and multiplicity of homoclinic solutions for a class of first-order periodic Hamiltonian systems with spectrum point zero are obtained. The proof is based on two critical point theorems for strongly indefinite functionals. Some recent results are improved and extended.