Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

Trading off Network Density with Frequency Spectrum for Resource Optimization in 5G Ultra-Dense Networks

To effectively increase the capacity in 5G wireless networks requires more spectrum and denser network deployments. However, due to the increasing network density, the coordination of network and spectrum management becomes a challenging task both within a single operator’s network and among multiple operators’ networks. In this article, we develop new radio resource management (RRM) algorithms for adapting the frequency spectrum and the density of active access nodes in 5G ultra-dense networks (UDNs) to the traffic load and the user density in different geographical areas of the network. To this end, we formulate a network optimization problem where the allocation of spectrum bandwidth and the density of active access nodes are optimized to minimize a joint cost function, and we exploit Lagrange duality techniques to develop provably optimal network-scheduling algorithms. In particular, we develop density algorithms for two application scenarios. The first scenario solves the resource management problem for an operator of an ultra-dense network with exclusive access to a pool of frequency resources, while the second scenario applies to the management of the network density of collocated UDNs that belong to multiple operators sharing the same frequency spectrum. Simulation results demonstrate how effectively the algorithms can adapt the allocation of the spectrum allocation and the density of active access nodes over space and time.

Inclusion properties of Generalized Integral Transform using Duality Techniques

Let Wβδ(α,γ) be the class of normalized analytic functions f defined in the region |z| < 1 and satisfyingRe eiθ((1-α+2γ)(f/z)δ+ (α-3γ+γ[(1-1/δ)(zf′/f) + 1/δ(1+zf″/f′)])(f/z)δ(zf′/f)-β)>0,with the conditions α ≤ 0, β < 1, γ ≤ 0, δ > 0 φ ∈ ℝ. For a non-negative and real- valued integrable function λ(t) with ∫01λ(t)dt = 1, the generalized non-linear integral transform is defined asVλδ(f)(z) = (∫01λ(t)(f(tz)/t)δdt)1/δ.The main aim of the present work is to find conditions on the related parameters such that Vλδ(f)(z) ∈ Wβ1δ1(α1,γ1) whenever f ∈ Wβ2δ2(α2,γ2). Further, several interesting applications for specific choices of λ(t) are discussed.

OPTIMAL ESTIMATES FOR THE SEMIDISCRETE GALERKIN METHOD APPLIED TO PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONSMOOTH DATA

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal$L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in$L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.